Module 5 Composition of Functions Homewor Score \( 3.25 / 16 \quad 5 / 16 \) answered If \( f(x)=x^{2}+2, g(x)=x-8 \), and \( h(x)=\sqrt{x} \), then \( (f \circ g)(x)= \)

Answer:

$20

Step-by-step explanation:

Given:

c=v+19

We are asked to find the cost when v=1

c=v+19

c=1+19

c=20

So, it will cost $20 for 1 visit.

Hope this helps! :)

Invariant points refer to the points that do not change their location on a graph or plane as the coordinates are subjected to some transformation.

A point (a, b) on a coordinate plane is said to be invariant under a function f if f(a, b) = (a, b).The invariant point(s) of the transformation that turns y = x² - 25 into y = √x²-25 are the point(s) that do not change their location on a graph or plane even after the coordinates are subjected to transformation. The transformation that turns y = x² - 25 into y = √x²-25 is the reflection over the x-axis, followed by the reflection over the y-axis. If the point (x, y) is invariant, then we will have:[tex]$$\sqrt{x^2 - 25} = y \ and \ y = x^2 - 25$$[/tex]Substituting y = x² - 25 in the first equation gives:[tex]$$\sqrt{x^2 - 25} = x^2 - 25$$$$x^4 - 50x^2 + 600 = 0$$[/tex]Solving for x gives:[tex]$$x = \pm \sqrt{10}, \pm 2\sqrt{5}$$[/tex]. Substituting x = ± √10 and x = ± 2√5 in the original equations gives the y-coordinates of the invariant points as:[tex]$$(\sqrt{10}, 5), \ (-\sqrt{10}, 5), \ (2\sqrt{5}, -15), \ (-2\sqrt{5}, -15)$$[/tex].

Therefore, the invariant points are:(√10, 5), (-√10, 5), (2√5, -15), and (-2√5, -15).

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The total number of bacteria present after 4 days is 12,800.

The total number of bacteria present after 4 days can be calculated using the formula: b(x) = 50 * 4^x, where x represents the number of days.

The given formula b(x) = 50 * 4^x represents the exponential growth of bacteria. The number 50 represents the initial number of bacteria, and 4 represents the number of bacteria doubles every 12 hours, which corresponds to a growth factor of 2 every 12 hours.

To find the total number of bacteria after 4 days, we substitute x = 4 into the formula:

b(4) = 50 * 4^4

    = 50 * 256

    = 12,800

Therefore, the total number of bacteria present after 4 days is 12,800.

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The average of all 12 numbers is 8(11)/12, which corresponds to option H.

To find the average of all 12 numbers, we can use the concept of weighted averages.

Given that the average of five numbers is 9, we can calculate the sum of those five numbers. Since the average is equal to the sum divided by the count, we have:

5 * 9 = 45

So, the sum of the first five numbers is 45.

Similarly, the average of seven other numbers is 8. Using the same logic, we can find the sum of those seven numbers:

7 * 8 = 56

The sum of the seven other numbers is 56.

To find the average of all 12 numbers, we need to calculate the total sum and divide it by the total count. The total sum is the sum of the first five numbers and the sum of the seven other numbers:

45 + 56 = 101

The total count is the sum of the counts of the two groups of numbers:

5 + 7 = 12

Now, we can find the average:

Average = Total Sum / Total Count = 101 / 12 = 8(11)/12

Therefore, the average of all 12 numbers is 8(11)/12, which corresponds to option H.

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The method used by Marie to solve the problem (-54 + 35) + (-35) = -54 + (35 + (-35)) is the associative property.

What is the associative property?

The associative property is a mathematical property of some binary operations that connect three or more elements. It involves changing the grouping of numbers being added or multiplied to get the same answer. This means that when grouping three or more quantities, the method used to group the addends or factors has no effect on the sum or product.

To clarify, in the expression a + (b + c) = (a + b) + c, the grouping of numbers b and c is changed. The associative property is said to be followed if the equation is valid.The method Marie used to solve the problem is given as follows: (-54 + 35) + (-35) = -54 + (35 + (-35)).

To solve, it involves adding (-54) and 35 first, followed by adding the sum to (-35).(-54 + 35) + (-35) = -19, on the left side.35 + (-35) = 0, on the right side,

Hence, the answer is:-54 + (35 + (-35)) = -54 + 0 = -54. Therefore, the method Marie used to solve the problem is the associative property.

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The quadratic function that contain the points are the correct answer is: f(x) = -3x² + 4x.

The standard form of a quadratic equation is given as: y = ax2 + bx + c Where, a, b and c are constants. Now, let us use the given coordinates and substitute the values of x and y into the standard form of the quadratic equation.

Then we can obtain a system of equations which can be used to solve for a, b and c. Let us substitute the first point (-1, -4).-4 = a(-1)2 + b(-1) + c … (1)

We also substitute the second point (0,0).0 = a(0)2 + b(0) + c... (2)Lastly, we substitute the third point (2,-10).-10 = a(2)2 + b(2) + c … (3)Now, we simplify these equations and solve for the coefficients. Let us start by simplifying equation (1):-4 = a - b + c

Also, equation (2) is simplified to:0 = c Finally, we simplify equation (3):-10 = 4a + 2b + c

By substituting 0 for c in equation (1), we get:-4 = a - b Then, we can solve for c by substituting a and b from equations (1) and (2) respectively. We get: c = -a + b - 4

Finally, we substitute c = 0 and solve for b in terms of a. We have:0 = -a + b - 4b = a + 4Thus, the quadratic function that passes through the points (-1,-4), (0,0) and (2,-10) can be obtained as: y = ax2 + bx + c, where a = -3, b = 4, and c = 0.

Substituting a, b and c values in the standard form of a quadratic function, we get: f(x) = -3x2 + 4x + 0Therefore, the correct answer is: f(x) = -3x² + 4x.

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The length of side b in the right triangle ABC, with A = 41° and c = 184 cm, is approximately 137 cm.

In the given right triangle ABC, with angle A = 41° and side c = 184 cm, we are asked to find the length of side b.

Using the trigonometric relationship in a right triangle, we can use the sine function:

sin(A) = b / c

Substituting the given values, we have:

sin(41°) = b / 184

To find the length of side b, we can rearrange the equation:

b = sin(41°) * 184

Using a calculator, we find that sin(41°) ≈ 0.6561.

Calculating b:

b = 0.6561 * 184

b ≈ 120.3436

Rounding to the nearest whole number, we find that the length of side b is approximately 137 cm.

Therefore, side b of the triangle has a length of approximately 137 cm.

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The expression 2sinθ(cscθ - sinθ) simplifies to 2cos²θ using the fundamental identities.

To simplify the expression 2sinθ(cscθ - sinθ) using the fundamental identities, we can start by using the reciprocal identity for cscθ:

cscθ = 1/sinθ

Substituting this into the expression, we have:

2sinθ(cscθ - sinθ) = 2sinθ(1/sinθ - sinθ)

Next, we can simplify the expression inside the parentheses by finding a common denominator:

2sinθ(1/sinθ - sinθ) = 2sinθ((1 - sin²θ)/sinθ)

Now, we can use the Pythagorean identity sin²θ + cos²θ = 1 to simplify the numerator (1 - sin²θ):

1 - sin²θ = cos²θ

Substituting this back into the expression, we have:

2sinθ((1 - sin²θ)/sinθ) = 2sinθ(cos²θ/sinθ)

Now, we can cancel out the sinθ terms:

2sinθ(cos²θ/sinθ) = 2cos²θ

Therefore, the expression 2sinθ(cscθ - sinθ) simplifies to 2cos²θ using the fundamental identities.

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1.  If Henry sells 3 computers in a day, he will earn $140.

2. Henry's earnings for selling 10 computers in a day would be $315.

To find Henry's daily earnings for selling 3 computers, we need to substitute x = 3 in the given function:

p(3) = 65 + 25(3)

p(3) = 65 + 75

p(3) = 140

Therefore, if Henry sells 3 computers in a day, he will earn $140.

Similarly, we can find his earnings for selling 4, 5, 6, 7, 8, 9, and 10 computers per day by substituting the respective values of x in the given function.

Number of computers sold (x) Earnings (p(x))

3 140

4 165

5 190

6 215

7 240

8 265

9 290

10 315

Thus, Henry's earnings for selling 10 computers in a day would be $315.

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The calculation using significant figures for 1.83 and 100.9 - 4.3 is as follows: When applying the rules of significant figures, it is important to consider the least precise measurement involved in the calculation. In this case, 1.83 has three significant figures, while both 100.9 and 4.3 have four significant figures.

When using significant figures, the main rule is to consider the least precise measurement involved in the calculation.

In the given calculation, 1.83 is a number with three significant figures, while 100.9 and 4.3 both have four significant figures. Since subtraction involves aligning the decimal places, the result will have the same number of decimal places as the measurement with the fewest decimal places. In this case, 1.83 has two decimal places, while 100.9 and 4.3 both have one decimal place. Therefore, the result will also have one decimal place.

Now let's perform the calculation:

100.9

-   4.3

= 96.6

The result of the subtraction, following the rules of significant figures, is 96.6. Since 96.6 has one decimal place, it matches the precision of the measurement with the fewest decimal places, which is 1.83.

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The probability that the sample proportion (the proportion living in the dormitories) is at least 0.70 is approximately 0.02. This is option A

Let p be the sample proportion living in the dormitories.The mean of the sample proportion is given by:μp = p = 0.60.

The standard deviation of the sample proportion is given by:σp = sqrt(p(1-p)/n)=sqrt(0.6*0.4/80)= 0.049.The sample size n = 80.

From Chebyshev' s theorem: P(|X - μ| ≥ k.σ) ≤ 1/k².

Substituting μ.p = 0.60 and σ.p = 0.049, we have:P(|p - 0.60| ≥ k*0.049) ≤ 1/k².

The question asks us to find the probability that the sample proportion (the proportion living in the dormitories) is at least 0.70.

So, we have:p ≥ 0.70 = 0.60 + k*0.049, k = (0.70 - 0.60)/0.049 = 2.04

.Substituting k = 2.04 in the above expression, we have:

P(|p - 0.60| ≥ 2.04*0.049) ≤ 1/(2.04)²= 0.2362.

So, P(p ≥ 0.70) = P(p - 0.60 ≥ 0.10)= P(p - 0.60/0.049 ≥ 2.04)= P(Z ≥ 2.04)≈ 0.0207.

Hence, the probability that the sample proportion (the proportion living in the dormitories) is at least 0.70 is approximately 0.02.

So, the correct answer is A

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To find f(8), we need to substitute x = 8 into the function. The value of f(8) is 71.

The given function f(x) has two parts, one for when x is less than or equal to 8, and another for when x is greater than 8. To find f(8), we substitute x = 8 into the function. Since x = 8 is equal to 8, which is not greater than 8, we use the first part of the function. Plugging in x = 8, we evaluate 8² + 7. Squaring 8 gives us 64, and adding 7 to it gives us 71. Therefore, the value of f(8) is 71. This means that when x is equal to 8, the function f(x) evaluates to 71. It is important to note that if x were greater than 8, we would use the second part of the function, -4x - 6, instead. However, in this case, x is not greater than 8, so we use the first part of the function to find f(8).

The function f(x) is defined as follows:

f(x) = {x² + 7 if x ≤ 8
        {-4x - 6 if x > 8

Since x = 8 is equal to 8, which is not greater than 8, we use the first part of the function.

Plugging in x = 8, we have:

f(8) = 8² + 7

Calculating this, we get:

f(8) = 64 + 7

f(8) = 71

Therefore, the value of f(8) is 71.

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The required answer is option (c), 0.01663 (3 significant figures) for the given operation: 0.79114.79×105. This is option C

The results of the given mathematical operations using the proper number of significant figures are:

a. 1.8427.2×15.63=441.666936 (5 significant figures)

b. 18.02×1.613.6=1.88 (3 significant figures)

c. 0.79114.79×105=0.01663 (3 significant figures)

d. 3.58×4.0=14.32 (3 significant figures)

e. 4.03.58=1.12 (3 significant figures)

f. 0.4511.4×10−4=3.15034965035×10^-5 (5 significant figures)

Hence, the required answer is option (c), 0.01663 (3 significant figures) for the given operation: 0.79114.79×105.

So, the correct answer is C

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[tex](8\cdot 320)^{\frac{1}{3}}\implies (2^3\cdot 320)^{\frac{1}{3}}\implies \left(2^3\cdot 2^6 \cdot 5 \right)^{\frac{1}{3}}\implies \left(2^{3+6} \cdot 5 \right)^{\frac{1}{3}} \\\\\\ \left(2^9 \cdot 5 \right)^{\frac{1}{3}}\implies 2^{9\cdot \frac{1}{3}}\cdot 5^{\frac{1}{3}}\implies 2^3\cdot 5^{\frac{1}{3}}\implies 8\sqrt[3]{5^1}\implies 8\sqrt[3]{5}[/tex]

The cost of gas was 3.051, the profit made by Mank is $366.96.

The amount of old copper tubing removed from air conditioning units is 68.8 kg.

The price paid per pound of copper tubing is $2.50.

The amount of gas used is 4 gallons.

The cost of 1 gallon of gas is $3.051.

By using dimensional analysis, we can convert 68.8 kilograms into pounds as shown below,

1 kg = 2.20462 lbs

68.8 kg = 68.8 × 2.20462 lbs= 151.663856 lbs

Using the above obtained value in pounds and the given price per pound, we can determine the total amount he was paid as shown below,

Total amount paid = 151.663856 × 2.5 = $379.16

The cost of 4 gallons of gas is 4 × 3.051 = $12.204

.Subtracting the gas cost from the amount paid, we get the profit,

Mank's profit = Total amount paid - Cost of gas= $379.16 - $12.204 = $366.956 or $366.96 (rounded to two decimal places)

Therefore, the profit made by Mank is $366.96.

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One statement that is necessarily true when a quadrilateral is inscribed in a circle is that the opposite angles of the quadrilateral are supplementary. In other words, the sum of the measures of two opposite angles is always 180 degrees.

To understand why this is true, let's consider a specific example. Imagine a quadrilateral PQRS inscribed in circle A. Let's say angle PQR measures 80 degrees. Since angle PQR is an inscribed angle that intercepts arc PS, it is equal in measure to half the measure of arc PS. Therefore, arc PS measures 160 degrees.

Now, let's focus on angle PSR. Angle PSR also intercepts arc PS, so it has the same measure as half the measure of arc PS, which is 160 degrees. Therefore, angle PSR measures 160 degrees as well.

If we add the measures of angles PQR and PSR, we get 80 + 160 = 240 degrees. This violates the fact that the sum of the measures of angles in a quadrilateral is always 360 degrees. Thus, it is clear that angles PQR and PSR cannot both measure 80 degrees.

Based on this example, we can conclude that the opposite angles of a quadrilateral inscribed in a circle are supplementary. This means that their sum is always 180 degrees.

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Answer:

Elena should eat half a serving of granola to consume 175 calories, based on her recipe.

Step-by-step explanation:

According to Elena's recipe, 5 servings of granola contain 1750 calories. This means that each serving of granola contains 1750 / 5 = 350 calories.

If Elena wants to eat 175 calories of granola, we need to find out how many servings this would be. We can do this by dividing the desired calorie amount (175 calories) by the calories per serving (350 calories).

So, 175 / 350 = 0.5 servings.

Therefore, Elena should eat half a serving of granola to consume 175 calories.

To graph the given piecewise function, we need to plot the two parts separately.

For x < -4, the graph is a straight line with a slope of -2 that extends indefinitely to the left.
For x ≥ -4, the graph is a curve that starts at the open dot (-4, 0) and moves upwards and to the right.

1. For x < -4:
  In this range, the function is defined as -2x. To graph this, we start at the point (-4, 8) and draw a line with a slope of -2. Since x is less than -4, the line will extend indefinitely to the left.

2. For x ≥ -4:
 In this range, the function is defined as √(x + 4). To graph this, we start at the point (-4, 0) and plot an open dot since the function is not defined at x = -4. Then, we draw a curve that represents the square root function. As x increases, the curve moves upwards and to the right.

Label the axis and mark the open dot at x = -4.

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The domain of the given function is, therefore,

[tex]\[ \left( -\infty ,-1 \right) \cup \left( -1,1 \right) \cup \left( 1,\infty  \right) \][/tex]

7.) Show that the following function is 1-1.

[tex]\[ f(x)=\frac{3 x-7}{2 x+1} \][/tex]

We know that if

[tex]\[f(a)=f(b) \Rightarrow a=b\][/tex]

then the function is said to be 1-1.

So, we need to show that the function given in the problem is 1-1.

Let us suppose that

[tex]\[\frac{3a-7}{2a+1}=\frac{3b-7}{2b+1}\][/tex]

Cross multiplying, we get

[tex]\[ (3a-7)(2b+1)=(3b-7)(2a+1) \][/tex]

Simplifying, we get

[tex]\[ 6ab-2a+3b-7=6ab-2b+3a-7 \][/tex]

Simplifying further,

[tex]\[5a-5b=0\]Thus,\[a=b\][/tex]

So, the function is 1-1.

8.) Find the domain of the following function.

[tex]\[ f(x)=\frac{\sqrt{x+7}}{x^{2}-1} \][/tex]

The denominator of the given function should not be zero.

Hence,[tex]\[x^{2}-1\neq 0\]\\[/tex]

Solving the above equation,[tex]\[x^{2}\neq 1\]or,\[(x+1)(x-1)\neq 0\]So,\[x+1\neq 0\]and \[x-1\neq 0\]or,\[x\neq 1\]and \[x\neq -1\][/tex]The domain of the given function is, therefore,

[tex]\[ \left( -\infty ,-1 \right) \cup \left( -1,1 \right) \cup \left( 1,\infty  \right) \][/tex]

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The monthly deposit Mona's parents need to start making when she is 7 is 109.42 / month.

How to calculate the monthly deposit to save $7500 by the time she reaches 13?

Savings Goal Calculator can be used to calculate this question.

Present Value of $7500 to be achieved at the age of 13 = $5,311.57.

Formula:Future Value = (Present Value) x (1 + i)^n

Where,i = rate of interest = 5% per annum

n = time period = 6 years (13 - 7)

Future Value = $7500 (as per question)

Present Value = ? = (Future Value) / (1 + i)^n= 7500 / (1 + 0.05)^6= 7500 / 1.3401

Present Value = $5,311.57

Let,Monthly Deposit = x

n = time period in months = 6 years (13 - 7) x 12 = 72

Formula:Monthly Deposit = (Future Value - Present Value) x i / [ (1 + i)^n - 1 ]

Monthly Deposit = (7500 - 5311.57) x 0.05 / [ (1 + 0.05)^72 - 1 ]

Monthly Deposit = 2188.43 x 0.05 / 7.759

Monthly Deposit = 109.42 / month

Therefore, the monthly deposit Mona's parents need to start making when she is 7 to have $7500 by the time she reaches the age of 13 is 109.42 / month.

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The solution to the initial value problem is[tex]\(y = 3e^{-7x}\).[/tex]

To solve this initial value problem, we'll use the method of separation of variables. The given differential equation is a first-order linear homogeneous differential equation. We can rearrange it as \(\frac{dy}{dx} = -7y\) and separate the variables by dividing both sides by \(y\):

[tex]\[\frac{1}{y} \, dy = -7 \, dx.\][/tex]

Next, we integrate both sides with respect to their respective variables. On the left side, we integrate \(\frac{1}{y}\,dy\) with respect to \(y\) and on the right side, we integrate \(-7\,dx\) with respect to \(x\):

[tex]\[\int \frac{1}{y} \, dy = \int -7 \, dx.\][/tex]

Integrating, we get \(\ln|y| = -7x + C\), where \(C\) is the constant of integration. Applying the initial condition \(y(\ln 5) = 3\), we substitute \(x = \ln 5\) and \(y = 3\) into the equation:

\[\ln|3| = -7(\ln 5) + C.\]

Simplifying, we find \(C = \ln|3| + 7(\ln 5)\). Substituting this back into the equation, we have:

\[\ln|y| = -7x + \ln|3| + 7(\ln 5).\]

Using the property of logarithms, we can combine the terms inside the logarithm:

[tex]\[\ln|y| = \ln|3 \cdot 5^7 e^{-7x}|\].[/tex]

Finally, using the fact that [tex]\(\ln e^x = x\)[/tex], we obtain:

[tex]\[\ln|y| = \ln|3 \cdot 5^7| - 7x.\[/tex]]

Removing the logarithms by taking the exponential of both sides, we get:

[tex]\[|y| = |3 \cdot 5^7|e^{-7x}.\][/tex]

Since \(y\) is a continuous function, we can remove the absolute value signs:

[tex]\[y = 3 \cdot 5^7e^{-7x}.\][/tex]

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The shortest distance from the ship to the shore, with two observation points 19 miles apart, is approximately 19 miles.

To find the shortest distance from the ship to the shore, we can use trigonometry and create a right triangle with the ship as the right angle.

Let's consider the observation point 55° north of East. We can draw a line from the ship to this point and label it as leg A. Similarly, for the observation point 35° south of East, we draw another line from the ship and label it as leg B.

Given that the two observation points are 19 miles apart, we have a triangle with side A = 19 miles and side B = 19 miles.

To find the shortest distance from the ship to the shore, we need to find the length of the hypotenuse, which represents the shortest distance. Let's label the hypotenuse as C.

Using trigonometric ratios, we can determine the lengths of sides A and B:

A = 19 * sin(55°)

B = 19 * sin(35°)

Finally, we can calculate the length of the hypotenuse C using the Pythagorean theorem:

C = sqrt(A^2 + B^2)

C ≈ sqrt(15.564^2 + 10.898^2)

C ≈ sqrt(242.235 + 118.765)

C ≈ sqrt(361)

C ≈ 19 miles

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You would need to maintain an average speed of 58 miles per hour to arrive at the concert on time.

You would need to maintain an average speed of approximately 20.55 miles per hour to arrive at school on time.

To calculate the average speed needed to arrive at the concert on time, you can use the formula:

Average speed = Total distance ÷ Total time

In this case, the total distance is 348 miles and the total time is 6 hours. So, the average speed required to reach the concert on time would be:

Average speed = 348 miles ÷ 6 hours = 58 miles per hour

Therefore, you would need to maintain an average speed of 58 miles per hour to arrive at the concert on time.

For the second question, to calculate the average speed needed to arrive at school on time, you can use the same formula:

Average speed = Total distance ÷ Total time

In this case, the total distance is 15 miles and the total time is 44 minutes. However, we need to convert the time to hours. Since there are 60 minutes in an hour, we divide 44 by 60 to get the time in hours: 44 minutes ÷ 60 = 0.73 hours.

Now, we can calculate the average speed:

Average speed = 15 miles ÷ 0.73 hours = 20.55 miles per hour

Therefore, you would need to maintain an average speed of approximately 20.55 miles per hour to arrive at school on time.

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594 mg is equivalent to 0.594 grams (g) or 0.000594 kilograms (kg).

To determine which quantities are equivalent to 594 mg, we need to convert milligrams to other units of mass.

1 gram (g) is equal to 1000 milligrams (mg). Therefore, dividing 594 mg by 1000 gives us 0.594 grams (g). So, 594 mg is equivalent to 0.594 g.

Similarly, 1 kilogram (kg) is equal to 1000 grams (g). Dividing 594 mg by 1000,000 (1000 x 1000) gives us 0.000594 kilograms (kg). Therefore, 594 mg is equivalent to 0.000594 kg.

In summary, 594 mg is equivalent to 0.594 grams (g) or 0.000594 kilograms (kg).

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In the number 0.3070 g, there are four significant figures.

To determine the significant figures, we follow these rules:

1. Non-zero digits are always significant. In this case, there are three non-zero digits: 3, 0, and 7.

2. Any zeros between non-zero digits are significant. Here, the zero between 3 and 7 is significant.

3. Leading zeros (zeros before the first non-zero digit) are not significant. In this case, the leading zero before 3 is not significant.

4. Trailing zeros (zeros after the last non-zero digit) that appear after a decimal point are significant. In this case, the trailing zero after 7 is significant because it appears after the decimal point.

Therefore, the number 0.3070 g has four significant figures: 3, 0, 7, and the trailing zero after 7.

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a) Show that any triangular region is convex:

A triangle is convex when any line segment linking any two points of the triangle is completely inside the triangle.

Let's show that a triangular region is convex by considering two points inside the triangular region and drawing a line segment between them and verifying that the line segment lies entirely inside the triangular region. Let's draw two points inside a triangular region, which we call A and B. Now let's draw a line segment that links A and B.

Let's consider three cases:

Case 1: The line segment lies entirely within the interior of the triangle, which means that we are done.

Case 2: The line segment intersects the boundary of the triangle exactly at a vertex. In this case, since A and B are in the interior of the triangle, they must be on different sides of the vertex. Thus, any point on the line segment between A and B lies on the interior of the triangle.

Case 3: The line segment intersects the boundary of the triangle along an edge, but not at a vertex. In this case, we can extend the line segment until it intersects the boundary at a vertex. Then we apply Case 2, which shows that the line segment lies entirely inside the triangular region.

b) Let T be a triangular region in the plane. Show that if P ∈ T and Q ∈ T then PQ must intersect one of the sides of T (perhaps at a vertex).

Proof:If PQ intersects the boundary of the triangle at a vertex, we are done. Thus, let us assume that PQ intersects the boundary of the triangle along a side, but not at a vertex. Without loss of generality, we can assume that PQ intersects the side BC between B and C.Let D be the intersection of PQ and the side AB. Then the triangle PBD is similar to the triangle ABC. Thus, the angle ∠PBD is equal to the angle ∠BAC. Similarly, the angle ∠PCD is equal to the angle ∠ABC. Thus, the sum of the angles ∠PBD and ∠PCD is equal to the sum of the angles ∠BAC and ∠ABC. However, the sum of the angles of a triangle is π radians (or 180 degrees). Thus, we haveπ = ∠PBD + ∠PCD + ∠ABC + ∠BAC > ∠ABC + ∠BAC,which implies that ∠ABC + ∠BAC < π. This is a contradiction since the sum of the angles of a triangle is π. Thus, PQ must intersect the boundary of the triangle at a vertex.

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The angular speed of the wheel is approximately 0.436 radians per second.

The linear speed in inches per second, we can use the formula:
Linear speed = radius × angular speed

The radius of the wheel is 26 inches and the wheel is rotating at 25 ∘ per second, the linear speed is :
Linear speed = 26 inches × 25 ∘ per second
= 650 inches per second
Therefore, the linear speed of the wheel is 650 inches per second.

For the angular speed in RPMs (Revolutions per minute), we need to convert the given angular speed from degrees per second to revolutions per minute.
Since there are 360 degrees in a revolution, we can calculate the angular speed in RPMs using the following formula:
Angular speed in RPMs = (Angular speed in degrees per second × 1 minute) / 360 degrees
The angular speed is 25 ∘ per second,the angular speed in RPMs is:
Angular speed in RPMs = (25 ∘ per second × 1 minute) / 360 degrees
= 25/360 RPM
= 0.069 RPM (rounded to three decimal places)
Therefore, the angular speed of the wheel is approximately 0.069 RPM.

For the angular speed in radians per second, we need to convert the given angular speed from degrees per second to radians per second.
Since there are 2π radians in a circle (360 degrees), we can calculate the angular speed in radians per second using the following formula:

Angular speed in radians per second = (Angular speed in degrees per second × 2π radians) / 360 degrees
The angular speed is 25 ∘ per second,the angular speed in radians per second is:
Angular speed in radians per second = (25 ∘ per second × 2π radians) / 360 degrees
= (25 × 2π) / 360 radians per second
= 0.436 radians per second (rounded to three decimal places)


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To find out how much more Tommy's account will be worth after 25 years, we need to calculate the future value of his account after 25 years at a 7.6% annual interest rate.

We can use the formula:

FV = PV x (1 + r)^n

where FV is the future value, PV is the present value, r is the annual interest rate, and n is the number of years.

Here, PV = $55,500 (annual contribution) x 25 (years) = $1,387,500. r = 7.6%, and n = 25.

So,FV = $1,387,500 x (1 + 0.076)^25= $186,514.

To find out how much John must contribute annually to retire at the same time as Tommy, we need to calculate the present value of his contributions and compare it to Tommy's account balance after 25 years, which is $186,514.

We can use the formula:PV = FV / (1 + r)^n

where PV is the present value, FV is the future value, r is the annual interest rate, and n is the number of years.

Here, FV = $1,500,000, r = 7.6%, and n = 25.

So,PV = $1,500,000 / (1 + 0.076)^25= $335,131.

John's present value after 25 years is $335,131.

He needs to contribute an additional amount to reach the goal of $1,500,000.

So,John's annual contribution = (goal - present value) / [(1 + r)^n - 1]= ($1,500,000 - $335,131) / [(1 + 0.076)^25 - 1]= $11,695.

To find out how long it will take John to reach the goal of $1,500,000, we can use the formula: n = log(FV/PV) / log(1 + r)

where FV is the future value, PV is the present value, r is the annual interest rate, and n is the number of years. Here, FV = $1,500,000, PV = $55,500 (annual contribution), r = 7.6%.

So,n = log($1,500,000 / $55,500) / log(1 + 0.076)= 36.95 years. years

.To find out how much sooner Tommy can retire, we need to calculate the number of years it will take him to reach $1,500,000 at a 9.6% annual interest rate.

We can use the formula:n = log(FV/PV) / log(1 + r)where FV is the future value, PV is the present value, r is the annual interest rate, and n is the number of years.

Here, FV = $1,500,000, PV = $55,500 (annual contribution), r = 9.6%.

So,n = log($1,500,000 / $55,500) / log(1 + 0.096)= 32.95 years. Tommy will retire 5 years sooner than John (37 years - 32 years = 5 years).

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The division of 2.079 by 0.693 is approximately equal to 2.996.

To divide 2.079 by 0.693, we follow these steps:

Write down the division problem: 2.079 ÷ 0.693.

Perform the division operation: Divide 2.079 by 0.693.

- Start by dividing the whole numbers: 2 ÷ 0 = 0.

- Bring down the decimal point and write 0. in the quotient.

- Now, divide the decimal parts: 0.079 ÷ 0.693.

- To simplify the division, multiply both the dividend and divisor by 1000 to eliminate the decimal points: 0.079 × 1000 ÷ 0.693 × 1000.

- This gives us 79 ÷ 693.

- Divide 79 by 693: 79 ÷ 693 = 0.114.

Combine the whole number and decimal part: 0 + 0.114 = 0.114.

Round the answer to the correct number of significant figures, which is three in this case. Since the digit following the third significant figure (4) is less than 5, we keep the third significant figure unchanged.

Therefore, the final answer is approximately 0.114.

Thus, 2.079 ÷ 0.693 is approximately equal to 2.996 when rounded to three significant figures.

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The new vector →u - →v is 〈 −10,5 〉. Hence, the correct option is (A) 〈 −10,5 〉.

Given vectors are →u = 〈 −6,−1 〉 and →v = 〈 4,−6 〉.We need to find the new vector →u − →v.Vector subtraction can be found by adding the additive inverse of the vector which we want to subtract. Therefore, the additive inverse of vector →v = 〈 4, −6 〉 is 〈 −4, 6 〉.Now, →u − →v  = →u + (−→v)  =  〈 −6,−1 〉 + 〈 −4,6 〉= 〈 −6−4,−1+6 〉 = 〈 −10,5 〉.Therefore, the new vector →u - →v is 〈 −10,5 〉. Hence, the correct option is (A) 〈 −10,5 〉.

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