atics For Senior High Schools lr Exercise 13.2 1. Simplify log 8 log 4 A 2. If log a = 2, log b = 3 and logc = -1, evaluate b 100ac (a) log. (b)log a³b the fall (c) log 2a√b 5c on a singla​

The slope of the tangent line to the curve at the point where t = 1 is 9.

To determine the slope of the tangent line to the curve defined by the parametric equations x(t) = 2t^3 - t^2 + 6t and y(t) = 9e^(6t - 6) at the point where t = 1, we can use the concept of differentiation.

First, let's find the derivative of x(t) and y(t) with respect to t:

dx(t)/dt = d/dt (2t^3 - t^2 + 6t)

= 6t^2 - 2t + 6

dy(t)/dt = d/dt (9e^(6t - 6))

= 54e^(6t - 6)

Next, we need to evaluate these derivatives at t = 1:

dx(1)/dt = 6(1)^2 - 2(1) + 6

= 6

dy(1)/dt = 54e^(6(1) - 6)

= 54e^0

= 54

Now, we have the slope of the tangent line at t = 1, which is given by dy(1)/dx(1). So, let's calculate that:

dy(1)/dx(1) = dy(1)/dt / dx(1)/dt

= 54 / 6

= 9

Therefore, the slope of the tangent line to the curve at the point where t = 1 is 9.

It's important to note that the slope represents the rate of change of y with respect to x at that specific point on the curve.

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The standard deviation for the given set of outcomes with their respective probabilities. The outcome probability of 0.19, 0.32, 0.15, 0.34 and 1234 2 is approximately 1128.96.

Calculate the expected value (mean) of the outcomes:

  Multiply each outcome by its corresponding probability.

  Sum up these products to obtain the expected value.

  In this case, the expected value is calculated as follows:

  (0.19 * 0) + (0.32 * 1) + (0.15 * 2) + (0.34 * 1234) + (2 * 2) = 250.28.

Calculate the squared difference between each outcome and the expected value:

   Subtract the expected value from each outcome.

  Square each of these differences.

  For each outcome, the squared difference from the expected value is calculated as follows:

 [tex](0 - 250.28)^2, (1 - 250.28)^2, (2 - 250.28)^2, (1234 - 250.28)^2, (2 - 250.28)^2.[/tex]

Multiply each squared difference by its corresponding probability:

  -Multiply each squared difference by the probability of the corresponding outcome.

  For each squared difference, multiply it by its corresponding probability:

  [tex](0 - 250.28)^2 * 0.19, (1 - 250.28)^2 * 0.32, (2 - 250.28)^2 * 0.15, (1234 - 250.28)^2 * 0.34, (2 - 250.28)^2 * 2.[/tex]

Sum up these products to obtain the variance:

  Add up the products obtained in the previous step.

  The variance is calculated as follows:

      [tex][(0 - 250.28)^2 * 0.19] + [(1 - 250.28)^2 * 0.32] + [(2 - 250.28)^2 * 0.15] + [(1234 - 250.28)^2 * 0.34] + [(2 - 250.28)^2 * 2] = 1272201.3524.[/tex]

Finally, calculate the standard deviation:

   Take the square root of the variance calculated in the previous step.

  The standard deviation is the square root of the variance:

  √(1272201.3524) ≈ 1128.96.

Therefore, the standard deviation of the given outcomes is approximately 1128.96.

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(a) A sequence (2n) of irrational numbers having a limit lim in that is a rational number:Consider the sequence (2n), where n is a positive integer. Here's the proof that this sequence converges to a limit, which is a rational number.

Observe that for every positive integer n, 2n can be written in terms of 2 as a power of 2, that is, 2n = 2^n. Since 2 is rational, so is 2^n. Therefore, (2n) is a sequence of irrational numbers having a limit that is a rational number, which is 0 when n approaches to negative infinity.(b) A sequence (rn) of rational numbers having a limit lim in that is an irrational number:Consider the sequence {rn} where rn = 1/n, n∈N.For every n∈N, rn is a rational number and lim (rn) = 0 which is an irrational number.

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The sequence 2, 2.8, 2.98, 2.998, 2.9998… is a sequence of irrational numbers which converges to a rational number 3.

The sequence (rn) is a sequence of rational numbers having a limit lim in that is an irrational number.

(a) A sequence (2n) of irrational numbers having a limit lim. In that is a rational number is:

There exist infinitely many sequences of irrational numbers, which converge to rational numbers.

Let us consider a sequence (2n) of irrational numbers, which converges to a rational number. 2, 2.8, 2.98, 2.998, 2.9998…

The sequence 2, 2.8, 2.98, 2.998, 2.9998… is a sequence of irrational numbers which converges to a rational number 3.

The limit of the sequence is 3, which is a rational number.

(b) A sequence (rn) of rational numbers having a limit lim in that is an irrational number:

One such example of a sequence (rn) of rational numbers having a limit lim in that is an irrational number is given below:

Consider the sequence (1 + 1/n)n, which is a sequence of rational numbers and converges to an irrational number e. The first few terms of the sequence are 2, 1.5, 1.33, 1.25, 1.2… and so on.

The limit of the sequence is e, which is an irrational number.

Thus, this sequence (rn) is a sequence of rational numbers having a limit lim in that is an irrational number.

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a. the conditional density of X given Y = y is 0, which means that X and Y are independent.

b.  the density function of Z = X + Y is:

f(Z) = d/dZ [f(V)]

= d/dZ [(1/2)e^(-2)V^2]

= (1/2)e^(-2)(Z^2)

(a)

To find the conditional density of X given Y = y, we use the formula:

f(X | Y = y) = f(X, Y)/f(Y)

where f(Y) is the marginal density function of Y.

First, we find the marginal density function of Y:

f(Y) = ∫ f(X, Y) dx (from x=0 to infinity)

= ∫ xe^(-2)(y+1) dx (from x=0 to infinity)

= e^(-2)(y+1) ∫ x dx (from x=0 to infinity)

= e^(-2)(y+1) [x^2/2] (from x=0 to infinity)

= infinity (since the integral diverges)

Since the integral diverges, we know that f(Y) cannot be a valid probability density function. However, we can still proceed to find the conditional density of X given Y = y:

f(X | Y = y) = f(X, Y)/f(Y)

= xe^(-2)(y+1) / infinity

= 0

So the conditional density of X given Y = y is 0, which means that X and Y are independent.

Similarly, to find the conditional density of Y given X = x, we use the formula:

f(Y | X = x) = f(X, Y)/f(X)

where f(X) is the marginal density function of X.

First, we find the marginal density function of X:

f(X) = ∫ f(X, Y) dy (from y=0 to infinity)

= ∫ xe^(-2)(y+1) dy (from y=0 to infinity)

= x/e^2 ∫ e^(-2)y dy (from y=0 to infinity)

= x/e^2 [e^(-2)y/-2] (from y=0 to infinity)

= xe^(-2)/2

Now we can find the conditional density of Y given X = x:

f(Y | X = x) = f(X, Y)/f(X)

= xe^(-2)(y+1)/[x e^(-2)/2]

= 2(y+1)/x

= 2/x * (y+1)

So the conditional density of Y given X = x is a function of y that depends on x.

(b)

To find the density function of Z = X + Y, we use the transformation method. We need to find the joint density function of U = X and V = X + Y, and then integrate over all possible values of U to get the marginal density function of V.

First, we need to find the inverse transformation functions:

X = U

Y = V - U

The Jacobian determinant of the transformation is:

J = |d(x,y)/d(u,v)| = |[∂x/∂u ∂x/∂v; ∂y/∂u ∂y/∂v]|

= |[1 0; -1 1]|

= 1

So the joint density function of U and V is:

f(U,V) = f(X,Y) * |J| = xe^(-2)(V-U+1)

We want to find the marginal density function of V:

f(V) = ∫ f(U,V) dU (from U=0 to V)

= ∫ xe^(-2)(V-U+1) dU (from U=0 to V)

= e^(-2)V ∫ x dx (from x=0 to V) + e^(-2) ∫ x dx (from x=V to infinity) + e^(-2) ∫ dx (from x=0 to V)

= e^(-2)V [V^2/2 - V^3/6] + e^(-2) [(x^2/2)] (from x=V to infinity) + e^(-2)V

= (1/2)e^(-2)V^3 - (1/6)e^(-2)V^3 + (1/2)e^(-2)V

+ (e^(-2)/2)(V^2 - 2V(V+1) + (V+1)^2) + e^(-2)V

= (1/2)e^(-2)V^2

So the density function of Z = X + Y is:

f(Z) = d/dZ [f(V)]

= d/dZ [(1/2)e^(-2)V^2]

= (1/2)e^(-2)(Z^2)

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In order to determine the values of x that meet the equation sin(3x) = -sin(x) on the interval [0, 277), we must first solve the sin(3x) equation.

We can proceed as follows using a calculator:

1. Enter sin(3x) = -sin(x) as the equation.

2. To isolate x, use the sine(-1) inverse function.

3. Find the value of x.

It's crucial to switch a calculator to radian mode before using it. After making the necessary computations, we discover that the equation's approximate solutions for the specified interval are:x ≈ 0, 1.57, 3.14, 4.71Consequently, the appropriate response isA. 0, 1.57, 3.14, 4.71

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The Secant method is a numerical method used to find the root of a mathematical equation. The method is based on the tangent line approximation of the function at a point.

To calculate the root of the function f(x) = x2 – 3x + 9 using the Secant method, perform the following steps:Step 1: Choose the initial guesses, co = 1, and x1 = 0. Step 2: Use the formula given below to compute the next approximation, xn+1:$$x_{n+1}=x_n-\frac{f(x_n)(x_n-x_{n-1})}{f(x_n)-f(x_{n-1})}$$ Step 3: Compute the approximate derivative (dfap) using the formula below:$$dfap=\frac{f(x_n)-f(x_{n-1})}{x_n-x_{n-1}}$$ Substituting the given values into the above equations, we have; $$f(x) = x^2 – 3x + 9$$$$c_0 = 1$$$$x_1 = 0$$$$x_2=x_1-\frac{f(x_1)(x_1-c_0)}{f(x_1)-f(c_0)}$$$$x_2=0-\frac{(0^2 - 3 \times 0 + 9)(0-1)}{(0^2 - 3 \times 0 + 9)- (1^2 - 3 \times 1 + 9)}$$$$x_2 = 1.5$$$$dfap = \frac{f(x_2) - f(x_1)}{x_2 - x_1}$$$$dfap = \frac{(1.5^2 - 3 \times 1.5 + 9) - (0^2 - 3 \times 0 + 9)}{1.5 - 0}$$$$dfap= -0.0033$$Hence, the approximate derivative (dfap) used in the first iteration of the Secant method is -0.0033.

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The approximate derivative (dfap) used in that iteration is 22.

The secant method is an iterative root-finding algorithm that utilizes a succession of roots of secant lines to better approximate a root of a function.

The secant method is a root-finding algorithm that doesn't require the function's derivative to be determined. This is a considerable advantage because computing derivatives can be difficult and can frequently take more time than computing a function value.

[tex]f(x) = x2 – 3x + 9[/tex]

To solve for the root of the function using secant method, we are given two initial guesses,

x0=1 and x1=0

Now, find the value of x2

The formula for calculating x2 is

[tex]x2 = x1 - f(x1)(x1-x0)/(f(x1)-f(x0))[/tex]

Now, we are given x0=1, x1=0

We need to calculate f(x0), f(x1) and dfap (approximate derivative)

First calculate f(x0) and f(x1)

[tex]f(x0) = x02 – 3x0 + 9f(1) \\= 1-3+9 \\= 7[/tex]

[tex]f(x1) = x12 – 3x1 + 9f(0) \\= 0-0+9 \\= 9[/tex]

So, using these values we calculate x2 which is

[tex]x2 = x1 - f(x1)(x1-x0)/(f(x1)-f(x0))\\= 0 - 9(0-1)/(9-7)\\= -9/2[/tex]

Next we calculate the approximate derivative, [tex]dfapdfap = f(x1)/dx[/tex]

Now, [tex]dx = (x2-x1) \\= (-9/2-0) \\= -9/2[/tex]

Therefore, [tex]dfap = f(x1)/dx\\= (f(x2) - f(x0))/(x2-x0)\\= ((-9/2)2 - 3(-9/2) + 9 - 7)/(-9/2-1) \\= 22[/tex]

So, the approximate derivative (dfap) used in that iteration is 22 (approx)

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The event that most contributed to the changing troop levels shown in the graph is when Communist troops launched a series of attacks during the Tet Offensive.

The Communist troops launched a series of attacks during the Tet Offensive to try to undermine American and South Vietnamese morale, cause a general uprising and seize control of the cities in South Vietnam.

However, this didn't go as planned, since the Communist troops suffered devastating losses on the battlefield.

The Tet Offensive, which was one of the most important turning points in the Vietnam War, led to changes in troop levels that are shown on the graph.

The Tet Offensive significantly increased troop levels because American forces had to respond with more soldiers and resources to defend against the attacks.

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The event which most contributed to the changing troop levels shown in the graph was the Communist troops launching a series of attacks during the Tet Offensive.

The Tet Offensive was a series of attacks on the cities and towns of South Vietnam by the People's Army of Vietnam (PAVN) (also known as the North Vietnamese Army or NVA) and the National Liberation Front of South Vietnam (NLF), commonly known as the Viet Cong.

The Tet Offensive began in the early hours of 30th January 1968, during the Vietnam War. This event had a significant impact on public opinion and led to the escalation of the war.The graph in question, which depicts the troop levels, demonstrates that there was a considerable rise in US troop numbers during the years leading up to the Tet Offensive.

Following this event, troop numbers rose even higher before declining in the years that followed.

Therefore, the Communist troops launching a series of attacks during the Tet Offensive contributed most to the changing troop levels shown in the graph.

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The Taylor series is a way to represent a function as a power series of its derivatives at a specific point in the domain. It is a crucial tool in calculus and its applications. The Taylor series for a function f(x) is given by:$$f(x) = \sum_{n=0}^\infty \frac{f^{(n)}(c)}{n!}(x-c)^n$$Where f^(n) (c) is the nth derivative of f evaluated at c.

In this case, we are asked to find the Taylor series centered at c=4 for the function f(x)=7sin(x).We first find the derivatives of f(x). The first four derivatives are:$f(x)=7sin(x)$;$f'(x)=7cos(x)$;$f''(x)=-7sin(x)$;$f'''(x)=-7cos(x)$;$f''''(x)=7sin(x)$;Notice that the pattern repeats after the fourth derivative. Thus, the nth derivative is:$f^{(n)}(x)=7sin(x+\frac{n\pi}{2})$Now, we can use the formula for the Taylor series and substitute in the derivatives evaluated at c=4:$f(x)=\sum_{n=0}^\infty \frac{7sin(4+\frac{n\pi}{2})}{n!}(x-4)^n$.

Thus, the Taylor series for f(x)=7sin(x) centered at c=4 is:$$7sin(x)=\sum_{n=0}^\infty \frac{7sin(4+\frac{n\pi}{2})}{n!}(x-4)^n$$.

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The values provided in the answer options (15.012, 10.526, and 25.851) are not the correct upper-tail critical value for the given scenario. The correct answer is 20.483.

To determine the upper-tail critical value for the chi-square (χ²) test with 10 degrees of freedom at a significance level of α = 0.025, we can refer to the chi-square distribution table or use statistical software. The correct upper-tail critical value for this test is approximately 20.483.

The chi-square distribution is a right-skewed distribution that is used in hypothesis testing to assess the association between categorical variables. The critical values of the chi-square distribution correspond to specific levels of significance and degrees of freedom.

In this case, we want to find the critical value for α = 0.025 (which corresponds to a two-tailed test with α/2 on each tail). With 10 degrees of freedom, we can consult a chi-square distribution table or use software to determine the critical value.

Using a chi-square distribution table, we look for the value that corresponds to the upper-tail area of 0.025 for 10 degrees of freedom. The critical value is the value that marks the boundary below which we reject the null hypothesis.

Based on the calculations, the upper-tail critical value for the chi-square test with 10 degrees of freedom and α = 0.025 is approximately 20.483. Therefore, any chi-square test statistic above this critical value would lead to the rejection of the null hypothesis at the specified level of significance.

It's important to note that the values provided in the answer options (15.012, 10.526, and 25.851) are not the correct upper-tail critical value for the given scenario. The correct answer is 20.483.

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The radius of convergence, R, of the series is 1. The interval of convergence, I, is (-1, 1) in interval notation.

The ratio test can be used to find the radius of convergence, R, of the given series. Applying the ratio test, we take the limit as n approaches infinity of the absolute value of the ratio of the (n+1)th term to the nth term. In this case, the (n+1)th term is [tex]((n+1)!x^{(n+1)})/(1.3.5....(2n+1))[/tex], and the nth term is [tex](n!x^n)/(1.3.5....(2n-1))[/tex].

Simplifying the ratio and taking the limit, we find that the limit is equal to the absolute value of x. Therefore, for the series to converge, the absolute value of x must be less than 1. This means that the radius of convergence, R, is 1.

To determine the interval of convergence, we need to find the values of x for which the series converges. Since the radius of convergence is 1, the series converges for values of x within a distance of 1 from the center of convergence, which is x = 0. Therefore, the interval of convergence, I, is (-1, 1) in interval notation.

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Out of the given options the true statements about the graph of             f(x) = cot(x) are B, C, and D.

B. f(x) is undefined for nπ, where n is an integer: The function cot(x) is defined as the ratio of cosine(x) to sine(x), which means it is undefined when sine(x) equals zero. This occurs at x = nπ, where n is an integer.

C. There is a vertical asymptote at x = π/2: As x approaches π/2, the value of cot(x) approaches positive infinity. Similarly, as x approaches -π/2, the value of cot(x) approaches negative infinity. This indicates the presence of vertical asymptotes at x = π/2 and x = -π/2.

D. F(x) is undefined when cos(x) = 0: The function cot(x) is undefined when cosine(x) equals zero. This happens at x = (2n + 1)π/2, where n is an integer.

A. (0,0) is a point on the graph: This statement is false. The value of cot(0) is undefined because it corresponds to dividing zero by zero, which is indeterminate.

E. All y-values are included in the range: This statement is false. The range of cot(x) is (-∞, -1) U (1, +∞), which means it does not include all possible y-values.

In conclusion, the true statements about the graph of f(x) = cot(x) are B, C, and D, while statements A and E are false.

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The correct option is (a) [7.53, 77.41].

To construct a 99% confidence interval for the true variance (σ²) of the increase in pulse rate of astronauts performing a given task, we can use the Chi-Square distribution.

The formula for the confidence interval for the variance is:

[ (n-1) * s² / χ²_upper , (n-1) * s² / χ²_lower ]

Where:

n is the sample size

s² is the sample variance

χ²_upper and χ²_lower are the upper and lower critical values from the Chi-Square distribution, respectively, based on the desired confidence level and degrees of freedom (n-1).

In this case, we have:

n = 12 (number of astronauts)

s² = (standard deviation)² = 4.28² = 18.2984

degrees of freedom = n - 1 = 12 - 1 = 11

critical values from the Chi-Square distribution for a 99% confidence level are χ²_upper = 26.759 and χ²_lower = 2.179

Now we can substitute these values into the formula to calculate the confidence interval:

[ (11 * 18.2984) / 26.759 , (11 * 18.2984) / 2.179 ]

Simplifying:

[ 7.531 , 77.414 ]

Therefore, the 99% confidence interval for the true variance (σ²) of the increase in the pulse rate of astronauts performing the given task is approximately [7.53, 77.41].

The correct option is (a) [7.53, 77.41].

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Answers of all logarithms are as follows a) log(27) + 2log(9) - log(54) can be expressed as log(81). b) log(12.5) + log(2) can be expressed as log(25). c) log(13.5) - log(10.5) can be expressed as log(1.285714286). d) log(64) + 2log(5) - 2log(40) can be expressed as log(25).

(a) We may use the properties of logarithms to express log(27) + 2log(9) - log(54) as a single logarithm. Let's dissect it step-by-step:

log(27) plus 2log(9) minus log(54)

= log(2187) - log(54) = log(2187/54), which equals log(81).

Thus, log(81) can be written as log(27) + 2log(9) - log(54).

(b) The addition property of logarithms can also be used to combine log(12.5) + log(2) into a single logarithm:

removing the amount we receive

In other words, log(12.5) + log(2) = log(25).

(c) We can apply the division property of logarithms to log(13.5) - log(10.5):

Log(13.5) - Log(10.5) = 13.5 - 10.5 = 1.285714286

Log(13.5) - log(10.5) is therefore equivalent to log(1.285714286).

(d) Finally, we may use the properties of logarithms to log(64) + 2log(5) - 2log(40):

log(64) = log(64) + 2log(5) - log(40)

= log(400) - log(16), log(400/16), log(400) - log(25), etc.

As a result, the equation log(64) + 2log(5) - 2log(40) can be written as

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The value of (a) is (−133 mod 23 261 mod 23) mod 23 equals 20 and the value of (b) is (457 mod 23 ⋅ 182 mod 23) mod 23 equals 16.

a) To calculate (−133 mod 23 261 mod 23) mod 23, we start by evaluating the innermost parentheses.

−133 mod 23 equals -10, because -133 divided by 23 gives a quotient of -5 with a remainder of -10.

Similarly, 261 mod 23 equals 7, because 261 divided by 23 gives a quotient of 11 with a remainder of 7.

Now, we substitute these values into the expression:

(-10 mod 23 7 mod 23) mod 23.

Next, we evaluate the outermost parentheses:

-10 mod 23 equals -10, and 7 mod 23 equals 7.

Finally, we substitute these values back into the expression:

(-10 mod 23 7 mod 23) mod 23 equals (-10 7) mod 23.

Calculating the subtraction first, we get -3 mod 23.

To ensure the result is positive, we add 23 to -3, giving us 20 mod 23.

Therefore, (−133 mod 23 261 mod 23) mod 23 equals 20.

b) To find (457 mod 23 ⋅ 182 mod 23) mod 23, we begin by evaluating the innermost parentheses.

457 mod 23 equals 4, as 457 divided by 23 gives a quotient of 19 with a remainder of 4.

Similarly, 182 mod 23 equals 4, because 182 divided by 23 gives a quotient of 7 with a remainder of 4.

Now, we substitute these values into the expression:

(4 ⋅ 4) mod 23.

Multiplying 4 by 4 gives us 16.

Finally, we substitute this value back into the expression:

(4 ⋅ 4) mod 23 equals 16 mod 23.

Therefore, (457 mod 23 ⋅ 182 mod 23) mod 23 equals 16.

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Answer:

[tex]\sf 3\:\dfrac{1}{5}\;miles[/tex]

Step-by-step explanation:

To find the total distance Eloise rides her bike, we need to add the distances she rode on Wednesday and Thursday.

First, convert the mixed numbers into improper fractions by multiplying the whole number by the denominator of the fraction, adding this to the numerator of the fraction, and placing the answer over the denominator.

[tex]\sf Wednesday: \quad 1 \frac{7}{10}\; miles=\dfrac{1 \cdot 10+7}{10}=\dfrac{17}{10}\; miles[/tex]

[tex]\sf Thursday: \quad 1 \frac{5}{10}\; miles=\dfrac{1 \cdot 10+5}{10}=\dfrac{15}{10}\; miles[/tex]

Add the two distances together.

As the denominators of the two fractions are the same, we simply add the numerators:

[tex]\sf \dfrac{17}{10}+\dfrac{15}{10}=\dfrac{17+15}{10}=\dfrac{32}{10}[/tex]

Simplify the improper fraction by dividing the numerator and denominator by 2:

[tex]\sf \dfrac{32 \div 2}{10 \div 2}=\dfrac{16}{5}[/tex]

Convert the improper fraction into a mixed number by dividing the numerator by the denominator:

[tex]\sf \dfrac{16}{5}=3\;remainder \;1[/tex]

The mixed number answer is the whole number and the remainder divided by the denominator:

[tex]3\frac{1}{5}[/tex]

Therefore, Eloise rides her bike a total of 3 1/5 miles.

Answer:

[tex]\mathrm{a=10,\ r=0.5}[/tex]

Step-by-step explanation:

[tex]\mathrm{The\ nth\ term\ of\ any\ geometric\ sequence\ is\ given\ by:}\\\mathrm{t_n=ar^{n-1}}\\\mathrm{Given,}\\\mathrm{5th\ term(t_5)=6.25}\\\mathrm{or,\ ar^{5-1}=6.25}\\\mathrm{or,\ ar^4=6.25......(1)}\\\\\mathrm{And,\ 7th\ term(t_7)=1.5625}\\\mathrm{or,\ ar^{7-1}=1.5625}\\\mathrm{or,\ ar^6=1.5625.........(2)}[/tex]

[tex]\mathrm{Dividing\ equation(2)\ by\ (1),}\\\mathrm{\frac{ar^6}{ar^4}=\frac{1.5625}{6.25}}\\\\\mathrm{or,\ r^2=\frac{1}{4}}\\\\\mathrm{or,\ r=\frac{1}{2}}[/tex]

[tex]\mathrm{From\ equation(1)\ we\ have}\\\mathrm{ar^4=6.25}\\\mathrm{or,\ a(0.5)^4=6.25}\\\mathrm{or,\ a=100}[/tex]

Alternative method:

[tex]\mathrm{Here,\ the\ sixth\ term\ of\ the\ sequence\ is\ geometric\ mean\ of\ the\ 5th\ and\ 7th}\\\mathrm{term.}\\\mathrm{So,\ we\ may\ say:}\\\mathrm{t_6=\sqrt{t_5\times t_7}}=\sqrt{6.25\times 1.5625}=3.125\\\mathrm{Now,\ common\ ratio(r)=\frac{t_6}{t_5}=\frac{3.125}{6.25}=\frac{1}{2}=0.5}\\\mathrm{We\ know,\ t_6=3.125}\\\mathrm{or,\ ar^5=3.125}\\\mathrm{or,\ a(0.5)^5=3.125}\\\mathrm{or,\ a=100}[/tex]

The first term and common ratio of the geometric progression (GP) can be determined based on given information. First term (a₁) is 100, and the common ratio (r) is ±0.5, leading to correct answer c. a₁ = 100, r = ±0.5.

By analyzing the values of the 5th and 7th terms, we can find the relationship between them and solve for the unknowns. The correct answer is c. a₁ = 100, r = ±0.5. In a geometric progression, each term is obtained by multiplying the previous term by a constant ratio. Let's denote the first term as a₁ and the common ratio as r. Based on the given information, the 5th term is 6.25 and the 7th term is 1.5625.

Using the formula for the nth term of a geometric progression, we can express these terms in terms of a₁ and r:

a₅ = a₁ * r⁴ = 6.25

a₇ = a₁ * r⁶ = 1.5625

To solve for a₁ and r, we can divide the equations:

(a₇ / a₅) = (a₁ * r⁶) / (a₁ * r⁴)

1.5625 / 6.25 = r²

0.25 = r²

Taking the square root of both sides, we have:

r = ±0.5 Substituting the value of r back into one of the equations, we can solve for a₁:

6.25 = a₁ * (0.5)⁴

6.25 = a₁ * 0.0625

a₁ = 6.25 / 0.0625

a₁ = 100

Therefore, the first term (a₁) is 100, and the common ratio (r) is ±0.5, leading to the correct answer c. a₁ = 100, r = ±0.5.

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i) Xi(f) = 5rect(f/2007)e^(-j2πft) | ii) Xz(f) = 5rect((f-400)/2007) | iii) X3(f) = 1 - 3*5rect(f/2007) + 1400(X(f) * X(f)) | iv) X(f) = 5rect(f/5)

we'll use properties of the Fourier transform and the given function x(t) = 5sinc(2007).

i) For xi(t) = -4x(t-4):

Using the time shifting property of the Fourier transform, we have:

Xi(f) = X(f)e^(-j2πft)

Since x(t) = 5sinc(2007), the Fourier transform X(f) of x(t) is given by:

X(f) = 5rect(f/2007)

Thus, substituting the values, we have:

Xi(f) = 5rect(f/2007)e^(-j2πft)

ii) For xz(t) = e^(j400)lx(t):

Using the frequency shifting property of the Fourier transform, we have:

Xz(f) = X(f - f0)

Since x(t) = 5sinc(2007), the Fourier transform X(f) of x(t) is given by:

X(f) = 5rect(f/2007)

Substituting the value f0 = 400, we have:

Xz(f) = 5rect((f-400)/2007)

iii) For X3(t) = 1 - 3x(t) + 1400xlx(t):

Using the linearity property of the Fourier transform, we have:

X3(f) = F{1} - 3F{x(t)} + 1400F{x(t)x(t)}

Since x(t) = 5sinc(2007), the Fourier transform X(f) of x(t) is given by:

X(f) = 5rect(f/2007)

Using the Fourier transform properties, we have:

F{x(t)x(t)} = X(f) * X(f)

Substituting the values, we have:

X3(f) = 1 - 3*5rect(f/2007) + 1400(X(f) * X(f))

iv) For X(t) = cos(400ft)x(t):

Using the modulation property of the Fourier transform, we have:

X(f) = (1/2)(X(f - 400f) + X(f + 400f))

Since x(t) = 5sinc(2007), the Fourier transform X(f) of x(t) is given by:

X(f) = 5rect(f/2007)

Substituting the value f = 400f, we have:

X(f) = 5rect((400f)/2007)

Simplifying, we have:

X(f) = 5rect(f/5)

To sketch the magnitude spectrum, Xo(f), we plot the magnitude of the Fourier transform for each case using the given formulas and the properties of the Fourier transform.

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The t procedure should not be used when there is a large outlier in the data or when the distribution shows moderate skewness. In these circumstances, the t procedure may not provide accurate results.

The t procedure assumes that the data is normally distributed. However, it can still be used under certain deviations from normality. The t procedure is robust to small departures from normality, so in the case of moderate skewness (option A), it can still provide reasonably accurate results. Skewness refers to the asymmetry of the distribution, and if it is only moderately skewed, the t procedure can be used.

However, there are situations where the t procedure should not be used. One such circumstance is when there is a large outlier in the data (option C). An outlier is an extreme value that differs significantly from the other observations. Large outliers can have a significant impact on the results of the t procedure, as it is sensitive to extreme values. In such cases, using the t procedure may lead to biased estimates or incorrect inferences.

Additionally, the sample standard deviation being large (option D) does not necessarily make the t procedure inappropriate. The t procedure is designed to handle variability in the data, including cases with larger standard deviations. As long as the other assumptions of the t procedure, such as normality and independence, are met, it can still be used effectively.

In summary, the t procedure should not be used when there is a large outlier in the data or when the distribution shows significant skewness. These situations can undermine the assumptions of the t procedure and may lead to inaccurate results.

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The function f(x) = x/(x + 6) does satisfy the hypothesis of the Mean Value Theorem on the given interval [1, 12].

To determine if the function satisfies the hypothesis of the Mean Value Theorem, we need to check two conditions: continuity and differentiability on the interval [1, 12].

Continuity: The function f(x) = x/(x + 6) is continuous on the interval [1, 12] because it is a rational function and the denominator (x + 6) is nonzero for all x in the interval.

Differentiability: The function f(x) = x/(x + 6) is differentiable on the interval (1, 12) since it is a quotient of two differentiable functions.

The derivative of f(x) can be calculated using the quotient rule, which yields f'(x) = 6/(x + 6)². The derivative is defined and nonzero for all x in the interval (1, 12).

Since the function is continuous on [1, 12] and differentiable on (1, 12), it satisfies the hypothesis of the Mean Value Theorem on the given interval.

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The probability that fewer than 60.96 patients will experience weight gain is approximately equal to 0.0274.

Given that, p = 0.14, q = 0.86 and n = 524

The number of successes for this problem (x) can range from 0 to 524.

Now, we can use the normal distribution formula below to approximate the probability:

P\left(x\leqslant z\right)=\frac{1}{\sqrt{2\pi}}\int_{-\infty}^{z} e^{-t^{2}/2}dt

Here, \mu = np = 524\cdot0.14 = 73.36 and \sigma =\sqrt{npq}= \sqrt{524\cdot0.14\cdot0.86}\approx6.50

Let x be the random variable and it follows a normal distribution with

\mu = 73.36 and \sigma =6.50.

Now, we can standardize the normal distribution using the formula z =\frac{x-\mu}{\sigma}.

Using this formula, we get z=\frac{60.96-73.36}{6.50}=-1.91

Putting this value of z in the above formula, we get: P(x<60.96)=P(z<-1.91)=0.0274

Therefore, the probability that fewer than 60.96 patients will experience weight gain is approximately equal to 0.0274.

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The shaded region represents the solution to the system of inequalities y ≥ 2x – 3 and y < 2x + 4.

Therefore, the second graph is correct.

To determine the solution to the system of inequalities y ≥ 2x – 3 and y < 2x + 4, we can start by graphing each inequality separately and then identifying the region that satisfies both conditions.

Let's graph the first inequality, y ≥ 2x – 3:

First, we'll plot the line y = 2x – 3. This line has a y-intercept of -3 and a slope of 2 (rise of 2 units for every 1 unit of horizontal movement).

Next, we'll determine which side of the line satisfies y ≥ 2x – 3. Since the inequality includes the "greater than or equal to" symbol, we'll shade the region above or on the line.

Now let's graph the second inequality, y < 2x + 4:

First, we'll plot the line y = 2x + 4. This line has a y-intercept of 4 and a slope of 2 (rise of 2 units for every 1 unit of horizontal movement).

Next, we'll determine which side of the line satisfies y < 2x + 4. Since the inequality includes the "less than" symbol, we'll shade the region below the line.

Now, we need to identify the region that satisfies both inequalities. This region is the overlapping area between the shaded regions of the two graphs.

Here's a visual representation of the solution [please refer to the graph added]

Hence, the shaded region represents the solution to the system of inequalities y ≥ 2x – 3 and y < 2x + 4.

Therefore, the second graph is correct.

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The probability that a randomly selected person who is tested positive is vaccinated is: 0.4895

We are given a two-way frequency table that represents the result of a recent study on the effectiveness of the flu vaccine.

The table is as follows:

                                 Pos.              Neg.                Total

Vaccinated                465                771                   1236

Not vaccinated         485                 600                 1085  

Total                           950               1371                   2321

Now we are asked to find the probability that a randomly selected person who tested positive for the flu is vaccinated.

Let A denote the event that the person is tested positive.

Let B denote the event that he/she is vaccinated.

A∩B denote the event that the person tested positive is vaccinated.

Let P denote the probability of an event.

We are asked to find:

P(B|A)

We know that:

P (B|A) = P (A∩B) / P (A)

Here,

P (A∩B) = 465 / 2321

And, P (A) = 950 / 2321

Hence,

P (B|A) = P (A∩B) / P (A)

P (B|A) = 465 / 950

P (B|A) = 0.4895

Therefore, The probability that a randomly selected person who is tested positive is vaccinated is: 0.4895

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The function is f(x,y)= 6xy. The region D is a triangle with vertices (0,0), (1,0), and (1,9).The region D can be represented by the limits 0 ≤ x ≤ 1 and 0 ≤ y ≤ 9x.

Therefore, the average value of f over D is given by:[tex]$$\bar f=\frac{\int_D f(x,y) dA}{\int_D dA}$$$$\int_D[/tex] [tex]f(x,y)dA= \int_{0}^{1}\int_{0}^{9x}6xydydx$$$$=\int_{0}^{1}3x(9x)^2dx$$$$=[/tex][tex]243/4$$[/tex]and the area of the region D is: $$\int_D dA = [tex]\int_{0}^{1}\int_{0}^{9x}dydx$$$$=\int_{0}^{1}9xdx$$$$=9/2$$[/tex]Therefore, the average value of f over D is[tex]:$$\bar f=\frac{\int_D f(x,y) dA}{\int_D dA}$$$$= \frac{243/4}{9/2}$$$$=27/2$$[/tex]Therefore, the average value of f over D is 27/2.

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The probability of Independent events P(A ∩ B) is 0.04.

Independent events are the events that do not influence the probability of each other when they occur simultaneously.

Thus, P(A ∩ B) can be calculated using the formula, P(A ∩ B) = P(A) × P(B).

Given that a and b are independent events with P(a) = 0.1 and P(b) = 0.4; hence;

P(A ∩ B) = P(A) × P(B)= 0.1 × 0.4= 0.04

Therefore, P(A ∩ B) = 0.04

Independent events occur when the occurrence of an event doesn't affect the occurrence of the other event.

To find P(A ∩ B), we multiply the probability of A with the probability of B.

P(A) = 0.1P(B) = 0.4

Now,P(A ∩ B) = P(A) * P(B)= 0.1 * 0.4= 0.04

Therefore, the probability of P(A ∩ B) is 0.04.

The definition of independent events and how to find P(A ∩ B). We multiply the probability of one event with the probability of the other event to find the probability of the intersection of two independent events.

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To find an objective function that has a maximum or minimum value at each indicated vertex, we need to consider the properties of the vertices.

Let's assume we have a set of vertices indicated by [tex]\(V = \{v_1, v_2, \ldots, v_n\}\).[/tex] To ensure that our objective function has either a maximum or minimum value at each vertex, we can construct a piecewise function that achieves this property.

First, we need to determine whether each vertex is a maximum or minimum point. Let's denote [tex]\(v_i\)[/tex] as a maximum vertex if the desired extremum at that vertex is a maximum value, and [tex]\(v_i\)[/tex] as a minimum vertex if the desired extremum is a minimum value.

For each vertex [tex]\(v_i\)[/tex], we can construct a quadratic function that achieves the desired extremum at that vertex. The general form of a quadratic function is [tex]\(f(x) = ax^2 + bx + c\).[/tex]

If [tex]\(v_i\)[/tex] is a maximum vertex, we choose a negative coefficient for the quadratic term [tex](\(a < 0\))[/tex] to ensure the function opens downwards and has a maximum value at that vertex. Conversely, if [tex]\(v_i\)[/tex] is a minimum vertex, we choose a positive coefficient for the quadratic term [tex](\(a > 0\))[/tex] to ensure the function opens upwards and has a minimum value at that vertex.

By assigning appropriate coefficients for each vertex, we can construct a piecewise function that satisfies the given conditions. The objective function can be defined as follows:

[tex]\[f(x) = \begin{cases} a_1 x^2 + b_1 x + c_1 & \text{if } x \in \text{Region 1} \\ a_2 x^2 + b_2 x + c_2 & \text{if } x \in \text{Region 2} \\ \ldots & \\ a_n x^2 + b_n x + c_n & \text{if } x \in \text{Region n} \end{cases}\][/tex]

Here, each region corresponds to a specific vertex [tex]\(v_i\)[/tex] and has its own set of coefficients ([tex]\(a_i, b_i, c_i\)[/tex]) chosen to achieve the desired maximum or minimum value at that vertex.

It's important to note that the specific regions and coefficients depend on the given vertices and their corresponding desired extremum values.

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The solution to the given differential equation [tex]\(xy' - y = y^2\)[/tex] that satisfies the initial condition (y(1) = -7) is (y = -7x).

To solve the given differential equation [tex](xy' - y = y^2)[/tex] with the initial condition (y(1) = -7), we can use the method of separable variables.

First, we rearrange the equation by dividing both sides by [tex]\(y^2\):[/tex]

[tex]\[\frac{xy'}{y^2} - \frac{1}{y} = 1\][/tex]

Now, we separate the variables and integrate both sides:

[tex]\[\int \frac{1}{y}\,dy = \int \frac{1}{x}\,dx + C\][/tex]

where (C) is the constant of integration.

Integrating the left side gives:

[tex]\[\ln|y| = \ln|x| + C\][/tex]

Next, we can simplify the equation by exponentiating both sides:

[tex]\[|y| = |x| \cdot e^C\][/tex]

Since (C) is an arbitrary constant, we can combine it with another constant,[tex]\(k = e^C\):[/tex]

[tex]\[|y| = k \cdot |x|\][/tex]

Now, we consider the initial condition (y(1) = -7). Substituting (x = 1) and (y = -7) into the equation, we get:

[tex]\[-7 = k \cdot 1\][/tex]

Therefore, (k = -7).

Finally, we can write the solution to the differential equation with the initial condition as:

[y = -7x]

where (x) can take any value except (x = 0) due to the absolute value in the solution.

The solution to the given differential equation that satisfies the initial condition (y(1) = -7) is (y = -7x).

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The equation relating A to B is A = 9 + B, and the graph is a line with a slope of 1 passing through the point (0, 9).

To write an equation relating A to B, we need to consider that Karen earns 9 dollars per hour working part-time at the bookstore.

Additionally, she earns one additional dollar for each book she sells.

Therefore, the equation relating A (the amount Karen earns in dollars) to B (the number of books she sells) can be expressed as:

This equation states that Karen's earnings in dollars (A) are equal to the base hourly wage of 9 dollars plus the additional earnings she receives for each book sold (B).

To graph this equation, we can plot the values on a coordinate plane. We'll assume B represents the horizontal axis (x-axis), and A represents the vertical axis (y-axis).

On the graph, the line starts at the point (0, 9) and has a slope of 1, indicating that for each additional book Karen sells, her earnings increase by 1 dollar.

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The new cost is C(3/14, 2/14, 1/14)*(Q+2)^(1/2) = (9/14)*(Q+2)^(1/2) pounds.

A firm's production quantity (q) is given by the expression, q(x,y,z)=(x²+y²+z²)^(1/2) where x, y, and z represent the number of kgs of copper, iron, and tin, respectively that it uses for production.

Given the cost of these metals, copper costing 1 pound per kg, iron 2 pounds per kg, and tin 3 pounds per kg, we have to use Lagrange multipliers to determine the amount of copper, iron, and tin required to minimize production costs if the firm needs to produce Q kg of its good.

The production cost (C) can be defined as follows:

C(x,y,z)=C_p(x)+C_i(y)+C_t(z) where C_p, C_i, and C_t are the costs of copper, iron, and tin, respectively, and they are given by: C_p(x)=1*x, C_i(y)=2*y, and C_t(z)=3*z.

Therefore, we can write the firm's production cost as C(x,y,z)=x+2y+3z.

Now we need to solve the following problem using the method of Lagrange multipliers:

minimize C(x,y,z)=x+2y+3z

subject to the constraint q(x,y,z)=(x²+y²+z²)^(1/2)=Q

where Q is the quantity the firm has to produce.

We need to set up the Lagrangian function: L(x,y,z,λ)=x+2y+3z-λ[(x²+y²+z²)^(1/2)-Q]

Then we find the partial derivatives of L with respect to x, y, z, and λ:

∂L/∂x=1-λx/[(x²+y²+z²)^(1/2)]∂L/∂y

=2-λy/[(x²+y²+z²)^(1/2)]∂L/∂z

=3-λz/[(x²+y²+z²)^(1/2)]∂L/∂λ

=(x²+y²+z²)^(1/2)-Q

=0

Now we solve the system of equations given by the partial derivatives and the constraint equation:

1-λx/[(x²+y²+z²)^(1/2)]=0 2-λy/[(x²+y²+z²)^(1/2)]

=0 3-λz/[(x²+y²+z²)^(1/2)]

=0 (x²+y²+z²)^(1/2)-Q

=0

From the first equation, we get:λx/(x²+y²+z²)^(1/2)=1, which means that λ=(x²+y²+z²)^(1/2)/x, or λ²(x²+y²+z²)=x², or λ²=(x/[(x²+y²+z²)^(1/2)])².

From the second equation, we get:λy/(x²+y²+z²)^(1/2)=2, which means that λ=2(y/[(x²+y²+z²)^(1/2)]), or λ²=(4y²)/(x²+y²+z²).

From the third equation, we get:λz/(x²+y²+z²)^(1/2)=3, which means that λ=3(z/[(x²+y²+z²)^(1/2)]), or λ²=(9z²)/(x²+y²+z²).

Now we can solve for x², y², and z² in terms of λ² by adding up the equations obtained from the second, third, and fourth equations:

x²+y²+z²=(1/λ²)(x²+y²+z²)[1+(4/9)+(1/4)]

=(14/9)(x²+y²+z²)/λ²x²+y²+z²

=(9/5)λ²

From the first equation, we have:

λ=±(x²+y²+z²)/(x²+y²+z²)^(1/2)

=(x²+y²+z²)^(1/2)

Using this value for λ, we can solve for x², y², and z².

We get:x²=(9/14)Q²y²=(4/14)Q²z²=(1/14)Q²

Now, we need to find the values of x, y, and z using these values of x², y², and z².

We get:

x=(3/14)Q^(1/2)y

=(2/14)Q^(1/2)z

=(1/14)Q^(1/2)

Therefore, the firm needs 3/14 kgs of copper, 2/14 kgs of iron, and 1/14 kgs of tin to produce the minimum amount of its good.

The minimum cost is given by C(3/14, 2/14, 1/14)

= (3/14) + 2*(2/14) + 3*(1/14)

= 9/14 pounds.

If the amount that needs to be produced increases by 2 kgs, then the new quantity is Q+2 kg.

Using the same process as before, we find that the new amounts of copper, iron, and tin required are (3/14)*(Q+2)^(1/2), (2/14)*(Q+2)^(1/2), and (1/14)*(Q+2)^(1/2).

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The relation 2 represents a function.

In order to determine which relation in the below table represents a function, we need to first understand what a function is.A function is a relationship in which each input value corresponds to exactly one output value.

To put it another way, each x-value has one and only one y-value. The most typical method to determine whether a relation is a function is to use the vertical line test.

The vertical line test is a way to determine if a relation is a function graphically. To test if a graph is a function, we draw a vertical line through each x-value on the graph. If a vertical line crosses the graph more than once, it is not a function.

If, on the other hand, the graph passes the vertical line test and no vertical line crosses the graph more than once, it is a function.Now let's look at the table below to determine which relation is a function.

We will first plot the x and y values of each relation on a coordinate system and then apply the vertical line test to each relation.

Relation 1: x | y0 | 10 | 11 | 22 | 23 | 34 | 35 | 4Relation 1 does not represent a function since we can draw a vertical line through x = 3 and the line will cross the graph more than once.

Relation 2: x | y2 | 33 | 34 | 45 | 46 | 57 | 5Relation 2 represents a function since we can draw a vertical line through each x-value on the graph and it will only cross the graph once.

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Arc ED is 72 Degrees-A)CED = 72 degrees ,B)ECD = 36 degrees ,C)CDE = 144 degrees, D)CAB = 90 degrees .E)DAB = 90 degrees

The measurements of the angles in the given scenario, we need to apply the properties of angles in a circle.

Given:

- Arc ED is 72 degrees.

- CD is the diameter of the circle.

Using the properties of angles formed by a chord and an arc, we can determine the measurements of the angles as follows:

A. CED:

The angle CED is formed by the arc ED. Since arc ED is given as 72 degrees, the measurement of angle CED is also 72 degrees.

B. ECD:

Angle ECD is an inscribed angle that intercepts arc ED. By the inscribed angle theorem, the measure of an inscribed angle is half the measure of the intercepted arc. Therefore, angle ECD is half of 72 degrees, which is 36 degrees.

C. CDE:

Angle CDE is formed by the chord CD. It is an opposite angle to angle ECD. Since the sum of opposite angles formed by a chord is always 180 degrees, angle CDE is also 180 - 36 = 144 degrees.

D. CAB:

Angle CAB is formed by the diameter CD. When a diameter of a circle creates an angle with any other point on the circle, the angle is always a right angle (90 degrees). Therefore, angle CAB is 90 degrees.

E. DAB:

Angle DAB is an inscribed angle that intercepts arc CD. Since CD is the diameter of the circle, the intercepted arc CD is a semicircle, which has a measure of 180 degrees. By the inscribed angle theorem, angle DAB is half of 180 degrees, which is 90 degrees.

To summarize:

A. CED = 72 degrees

B. ECD = 36 degrees

C. CDE = 144 degrees

D. CAB = 90 degrees

E. DAB = 90 degrees

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