Find the values of the six trigonometric functions if the conditions provided hold.cos(2θ) = 1/sqrt290° ≤ θ ≤ 180 and

The approximation provides a second-order accurate estimate of the derivative f'(x) using the function values f(x), f(x + h), and f(x + 2h).

To derive a second-order accurate, one-sided difference approximation to f'(x) using Taylor series, we can start by expanding the function f(x + h) and f(x + 2h) in their Taylor series expansions around x.

Using Taylor series expansion for f(x + h), we have:

f(x + h) = f(x) + f'(x)h + f''(x)(h²)/2 + O(h³)

Using Taylor series expansion for f(x + 2h), we have:

f(x + 2h) = f(x) + 2hf'(x) + 2h²f''(x) + O(h³)

Now, let's construct a one-sided difference approximation for f'(x) using these expansions.

Taking the difference between f(x + h) and f(x), we get:

f(x + h) - f(x) = f'(x)h + f''(x)(h²)/2 + O(h³)

Similarly, taking the difference between f(x + 2h) and f(x + h), we get:

f(x + 2h) - f(x + h) = f'(x)h + f''(x)h² + O(h³)

We can rearrange the first equation to solve for f'(x):

f'(x) = (f(x + h) - f(x))/h - f''(x)(h/2) + O(h²)

Substituting the second equation into the above expression, we have:

f'(x) = (f(x + h) - f(x))/h - (f(x + 2h) - f(x + h))/(2h) + O(h²)

Simplifying the expression, we get the second-order accurate, one-sided difference approximation to f'(x):

f'(x) ≈ (3f(x + h) - 4f(x) + f(x + 2h))/(2h)

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The graph will have a positive slope, indicating an increasing number of cakes made over time. It will start at (0, 10) and continue with a Straight line upwards as time progresses.

The graph of the relationship between the number of cakes made and time, we can use a coordinate plane where the x-axis represents time (in hours) and the y-axis represents the number of cakes made.

Since the baker can make the same number of cakes each hour, we know that the rate of cake production is constant. Therefore, the graph will be a straight line with a constant slope.

Given that the baker has already made 10 cakes, we can start the graph at the point (0, 10) on the coordinate plane. This represents the initial time (0 hours) and the initial number of cakes (10).

Next, we can plot additional points on the graph using the information that the baker makes the same number of cakes each hour for 5 hours. Since the rate is constant, we can add the same value to the y-coordinate for each point.

For example, after 1 hour, the baker would have made 10 cakes + 1 cake (assuming she can make one cake per hour), resulting in the point (1, 11). Similarly, after 2 hours, the baker would have made 10 cakes + 2 cakes, resulting in the point (2, 12), and so on.

Connecting these points with a straight line will give us the graph of the relationship between the number of cakes made and time.

The graph will have a positive slope, indicating an increasing number of cakes made over time. It will start at (0, 10) and continue with a straight line upwards as time progresses.

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An example of a condition that has a specialty growth chart is Turner syndrome.

Turner syndrome is a genetic condition in which a female is born with only one X chromosome or partially missing X chromosome. It is associated with specific growth patterns and may result in shorter stature.

A specialty growth chart for Turner syndrome takes into account these unique growth patterns and helps monitor growth and development in affected individuals. The growth chart for Turner syndrome is tailored to the condition, considering factors such as age, bone age, and growth hormone therapy if applicable.

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The bus has a higher rate of motion or Velocity.

The vehicle travels the fastest,compare their respective speeds. Speed is defined as the distance traveled per unit of time.

the speed of the bus and the car:

Speed of the bus = Distance / Time = 5 kilometers / 10 minutes = 0.5 kilometers per minute.

Speed of the car = Distance / Time = 9 kilometers / 20 minutes = 0.45 kilometers per minute.

Comparing the speeds, we can see that the bus travels at a speed of 0.5 kilometers per minute, while the car travels at a speed of 0.45 kilometers per minute.

Therefore, the bus travels faster than the car. It covers a greater distance in the same amount of time compared to the car.

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Answer:

Therefore, the median for the given data set is 28.

Step-by-step explanation:

To find the median for the given data set, you need to arrange the numbers in ascending order and determine the middle value.

Arranged data set: 8, 9, 19, 28, 33, 37, 46

Since there is an odd number of values (7 in this case), the median will be the middle value.

Median = 28

Therefore, the median for the given data set is 28.

Answer:

£3/£8 = 37.5/100 = 3/8.

Step-by-step explanation:

To write £3 as a fraction of £8, we can use the following formula:

Part/Whole = Percent/100

In this case, the whole is £8 and the part is £3, so we have:

£3/£8 = x/100

To solve for x, we can cross-multiply:

£3 * 100 = £8 * x

300 = 8x

x = 300/8

x = 37.5

Therefore, £3 is equivalent to 37.5% of £8. We can also write this as a fraction:

£3/£8 = 37.5/100 = 3/8.

The angular momentum vector of the rotating disk is:

L = (0.0002 kg * m²) * ω * k

Momentum is characterised as the intensity of a body's motion. As momentum depends on both velocity and the direction of the body's motion, it is quantified by "mass velocity".

To find the angular momentum vector of a rotating disk, we need to know the angular velocity and the moment of inertia of the disk.

Given:

- Mass of the disk (m): 2.0 kg

- Diameter of the disk (d): 4.0 cm

First, let's convert the diameter to meters:

- Diameter (d) = 4.0 cm = 0.04 m

The moment of inertia of a disk can be calculated using the formula:

I = (1/4) * m * r²

where m is the mass and r is the radius of the disk.

The radius of the disk is half of the diameter:

- Radius (r) = d/2 = 0.04 m / 2 = 0.02 m

Now we can calculate the moment of inertia:

I = (1/4) * 2.0 kg * (0.02 m)² = 0.0002 kg * m²

Next, let's assume the disk is rotating about its central axis with an angular velocity of ω.

The angular momentum vector (L) of the disk is given by the formula:

L = I * ω

Since the angular momentum is a vector, we need to specify its direction using unit vectors. In this case, since the disk is rotating about its central axis, the angular momentum vector is along the axis perpendicular to the plane of the disk.

Therefore, the angular momentum vector of the rotating disk can be written as:

L = I * ω * k

where k is the unit vector along the z-axis (perpendicular to the plane of the disk).

Note: The magnitude of the angular momentum vector can be obtained by taking the cross product of the moment of inertia vector and the angular velocity vector.

So, the angular momentum vector of the rotating disk is:

L = (0.0002 kg * m²) * ω * k

Please note that the value of the angular velocity (ω) is needed to compute the exact angular momentum vector.

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Answer:

See below, answer is 30

Step-by-step explanation:

3(8+2) = 3(8) + 3(2) = 24 + 6 = 30

You can also do 3(8+2) = 3(10) = 30

The probability that at least one person is vaccinated is 1 - (0.32), which is equal to 0.68.

To calculate the probability that at least one out of four randomly selected people has been vaccinated, we can use the complementary probability approach. The complementary probability is equal to 1 minus the probability of the event not occurring.

Given that 68% of the population has been vaccinated, the probability that an individual has been vaccinated is 0.68. Therefore, the probability that an individual has not been vaccinated is 1 - 0.68, which is 0.32.

Using the hint provided, we can calculate the probability that none of the four people have been vaccinated as (0.32) raised to the power of 4 since the events are independent and the probability of each event is 0.32.

So, the probability that at least one person is vaccinated is equal to 1 minus the probability that none of the four people are vaccinated. Thus, the probability is 1 - (0.32) = 0.68.

In summary, the probability that at least one out of four randomly selected people has been vaccinated is 0.68, given that 68% of the population has been vaccinated. This is calculated by subtracting the probability that none of the four people are vaccinated from 1.

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Answer: D.

Step-by-step explanation:

In triangle ∆ABC, the ratio that equals cos C is (c - a)

(a) The Taylor polynomial of degree 4 for f(x) about the point x = 1 is P4(x) = (x - 1) - (x - 1)^2/2 + (x - 1)^3/3 - (x - 1)^4/4. (b) Approximations for f(0.9) and f(1.1) using the Taylor polynomial P4(x) are f(0.9) ≈ -0.105 and f(1.1) ≈ 0.095. (c) The error bound for both approximations f(0.9) and f(1.1) using the Taylor polynomial P4(x) is approximately 0.00012.

a) To find the Taylor polynomial of degree 4 for f(x) about x = 1, we'll use the formula for the Taylor series expansion:

Pn(x) = f(a) + f'(a)(x - a) + f''(a)(x - a)^2/2! + f'''(a)(x - a)^3/3! + ... + fⁿ(a)(x - a)^n/n!

For the given function f(x) = ln(x), let's calculate the derivatives up to the fourth order:

f(x) = ln(x)

f'(x) = 1/x

f''(x) = -1/x²

f'''(x) = 2/x³

f⁴(x) = -6/x⁴

Now, substitute x = 1 and a = 1 into the formula to get the Taylor polynomial of degree 4:

P4(x) = ln(1) + (1/1)(x - 1) + (-1/1²)(x - 1)²/2! + (2/1³)(x - 1)³/3! + (-6/1⁴)(x - 1)⁴/4!

Simplifying the terms, we get:

P4(x) = (x - 1) - (x - 1)^2/2 + (x - 1)^3/3 - (x - 1)^4/4

(b) To approximate f(0.9) and f(1.1) using the Taylor polynomial P4(x), we substitute the respective values of x into P4(x):

For f(0.9):

f(0.9) ≈ P4(0.9)

       = (0.9 - 1) - (0.9 - 1)^2/2 + (0.9 - 1)^3/3 - (0.9 - 1)^4/4

Calculating the expression gives f(0.9) ≈ -0.105.

Similarly, for f(1.1):

f(1.1) ≈ P4(1.1)

       = (1.1 - 1) - (1.1 - 1)^2/2 + (1.1 - 1)^3/3 - (1.1 - 1)^4/4

Calculating the expression gives f(1.1) ≈ 0.095.

(c) The Taylor remainder formula allows us to estimate the error between the actual function and its Taylor polynomial approximation. For the Taylor polynomial P4(x), the remainder term R4(x) is given by:

R4(x) = (x - a)⁵/f⁵(c)(5!)

Where a = 1 (the point of expansion) and c is some value between 1 and x.

To find the error bound, we need to evaluate the fifth derivative of f(x) = ln(x):

f⁵(x) = 24/x⁶

To find the maximum value of f⁵(c) for c between 1 and x, we consider the interval [0.9, 1.1]. The maximum value occurs at x = 0.9:

f⁵(c) = 24/0.9⁶

Calculating this expression, we find that f⁵(c) ≈ 379.08.

Now, substituting the values into the remainder formula, we have:

R4(x) = (x - 1)⁵/(379.08)(5!)

For both f(0.9) and f(1.1), the error bound for both approximations f(0.9) and f(1.1) using the Taylor polynomial P4(x) is approximately 0.00012.

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Therefore, the rectangular coordinates for point P are (2.828, 2.828).

Rectangular coordinates are used to locate points on the coordinate plane. A point P can be represented using rectangular coordinates as (x, y). Let us look at an example to understand this concept better:Example:Let us consider the point P that lies on the coordinate plane.

If the point P has a distance of 4 units from the origin and an angle of 45 degrees with the positive x-axis, what are the rectangular coordinates, (x, y) for P?Solution:We know that the rectangular coordinates for a point P can be represented as (x, y).

We are given that the point P has a distance of 4 units from the origin and an angle of 45 degrees with the positive x-axis.Using this information, we can find the value of x and y.

Using trigonometry, we know that x = r cos(theta) and y = r sin(theta), where r is the distance from the origin to the point P, and theta is the angle that the line segment connecting the origin to the point P makes with the positive x-axis.In this case, r = 4 and theta = 45 degrees.Substituting these values in the formula for x and y, we get

:x = r cos(theta) = 4 cos(45) = 4 * (1/sqrt(2)) = 2.828y = r sin(theta) = 4 sin(45) = 4 * (1/sqrt(2)) = 2.828

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It is unlikely that the sample of cars with a mean weight of 1976 Kg can be regarded as a sample from a large population of cars with a mean weight of 1500 Kg and a standard deviation of 130 Kg.

To determine if the sample can be reasonably regarded as a representative sample of the population, a hypothesis test can be performed. The null hypothesis (H0) assumes that the sample is from the population with a mean weight of 1500 Kg, while the alternative hypothesis (H1) assumes that the sample is not from the population.

Using the given information, we can calculate the test statistic, which is the z-score in this case. The z-score is calculated as (sample mean - population mean) / (population standard deviation/sqrt (sample size)). Comparing the obtained z-score to the critical value at the 5% level of significance (typically obtained from a standard normal distribution table), we can determine if the null hypothesis should be rejected.

If the calculated z-score exceeds the critical value, we reject the null hypothesis and conclude that the sample is not likely to be from the specified population. However, if the calculated z-score is within the acceptance range, we fail to reject the null hypothesis, indicating that the sample is reasonably representative of the population.

Therefore, by performing the hypothesis test at a 5% level of significance, we can determine the likelihood of the sample cars with a mean weight of 1976 Kg being a representative sample from a population with a mean weight of 1500 Kg and standard deviation of 130 Kg.

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Answer:

ti is 20

Step-by-step explanation:

ti's cuz I got 52 and I did was - 52 with 32 and I got 20

(1,2,4)

Range describes the y-values of a graph.

Range

Range is the y-values that a graph covers. Remember that the y-values are found on the vertical axis. If the graph is not continuous, then the values between the points are not included in the range. Similar to the range, the domain of a graph is the x-values that a graph covers. If there is a coordinate point with a y-value, then that y-value should be included in the range.

Finding Range

In order to find the range, we need to find all the unique y-values of the graph. Additionally, the range is given in numerical order. This means starting from the least value and going up to the greatest. The lowest y-value is 1, then 2, and finally 4. Even though there are two points where y = 2, we are only looking for unique values. This means that the range is (1,2,4).

A Pearson's r correlation coefficient of .71 for age and attention span would be an example of a strong positive correlation.

The Pearson correlation coefficient, denoted by "r," is a statistical measure that quantifies the strength and direction of the linear relationship between two variables. The values of r range from -1 to +1.

A positive correlation indicates that as one variable (in this case, age) increases, the other variable (attention span) also tends to increase. The closer the value of r is to +1, the stronger the positive correlation.

In this example, a correlation coefficient of .71 indicates a relatively strong positive relationship between age and attention span. It suggests that as individuals get older, their attention span tends to increase.

However, it's important to note that correlation does not imply causation. While there is a strong association between age and attention span in this case, other factors could also influence attention span independently of age.

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Answer: y = $50x + $25

Step-by-step explanation:

      First, let x be equal to the hours of service and y be the total cost. Since the cost is 50 dollars per hour of service, we will write this as "y = 50x" for our equation.

      Next, there is a 25-dollar charge for each call. We will add this to our equation as "y = 50x + 25."

Our equation is;

      y = $50x + $25

To find the closed form of the power series, we need to determine the explicit formula for the terms of the geometric series and then express it as a power series.

The geometric series is given by ∑n=0 [infinity] xn, where x is a constant.

The explicit formula for the terms of the geometric series is given by xn = x^n.

Now, let's express the geometric series as a power series:

∑n=0 [infinity] xn = ∑n=0 [infinity] x^n

To express this as a power series, we need to rewrite it in terms of the variable t, where t = x^n.

We can rewrite x^n as (x^1)^n = (x)^n.

Now, our series becomes:

∑n=0 [infinity] (x)^n

This is a geometric series with a common ratio of x. In order for the series to converge, the absolute value of x must be less than 1 (|x| < 1).

The formula for the sum of a convergent geometric series is:

S = a / (1 - r)

where a is the first term and r is the common ratio.

In this case, the first term (a) is 1 and the common ratio (r) is x.

So, the closed form of the power series is:

S = 1 / (1 - x)

Therefore, for |x| < 1, the closed form of the power series is 1 / (1 - x).

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The area of the shaded part is 45 cm² .

Given figure is divided into two triangles of same bases but of different heights. Calculate the areas of bigger triangle( outer one ) and unshaded triangle.

Area of triangle = 1/2×b×h

b = base of triangle

h = height of triangle

Firstly calculate the area of bigger triangle with dimensions,

b= 18 cm

h= 9 cm

Area of triangle = 1/2×b×h

                          = 1/2×18×9

                          = 81 cm²

Now calculate the area of unshaded triangle with dimensions,

b= 18 cm

h= 4 cm

Area of triangle = 1/2×b×h

                          = 1/2×18×4

                          = 36 cm²

Now to calculate the area of the shaded region,

Area of shaded region = Total area of triangle( bigger triangle ) - Area of unshaded triangle.

Area of shaded region = 81 cm² - 36 cm²

Area of shaded region = 45 cm²

Hence area of shaded part of the figure is 45 cm².

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Answer:

on the TI-30XS Multiview calculator you click on the 2nd button first then you click on the x² button to make the square root and lastly you just put any number on it to give you the answer.

Step-by-step explanation:

isolate the squared term and the constant term on opposite sides of the equation. Then take the square root of both sides, making the side with the constant term plus or minus the square root.

a) {0000000, 1111111}: Can detect 6 errors and correct 3 errors.

b) {00000, 00111, 10101, 11011}: Can detect 1 error and cannot correct any errors.

c) {00000000, 11111000, 01100111, 100101101}: Can detect 2 errors and correct 1 error

To determine the number of errors each binary code can detect and correct, we need to analyze the Hamming distance of the codes.

The Hamming distance is the number of positions at which two code words differ.

a) {0000000, 1111111}:

Since there are only two code words, the minimum Hamming distance is the number of differing positions between the two code words. In this case, the Hamming distance is 7 since all positions differ.

Number of errors detectable: (Hamming distance - 1) = 7 - 1 = 6 errors

Number of errors correctable: (Hamming distance - 1) / 2 = 6 / 2 = 3 errors

Therefore, this code can detect up to 6 errors and correct up to 3 errors.

b) {00000, 00111, 10101, 11011}:

In this code, the minimum Hamming distance is 2 since there are pairs of code words that differ in exactly two positions.

Number of errors detectable: (Hamming distance - 1) = 2 - 1 = 1 error

Number of errors correctable: (Hamming distance - 1) / 2 = 1 / 2 = 0 errors

Therefore, this code can detect 1 error but cannot correct any errors.

c) {00000000, 11111000, 01100111, 100101101}:

The minimum Hamming distance in this code is 3 since there are pairs of code words that differ in exactly three positions.

Number of errors detectable: (Hamming distance - 1) = 3 - 1 = 2 errors

Number of errors correctable: (Hamming distance - 1) / 2 = 2 / 2 = 1 error

Therefore, this code can detect up to 2 errors and correct up to 1 error.

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The first 5 nonzero terms of the power series are: (1/7)x²7 - (1/51)(x²17)/3! + (1/155)(x²27)/5! - (1/315)(x²37)/7! + (1/561)(x²47)/9! + ...

To find the power series representation of the indefinite integral ∫x²6×sin(x²5) dx, we can use the power series expansion of the sine function and integrate each term.

The power series expansion of sin(x) is given by:

sin(x) = x - (x²3)/3! + (x²5)/5! - (x²7)/7! + ...

To find the power series representation of the given integral, we substitute x²5 for x in the power series expansion of sin(x):

sin(x²5) = x²5 - (x²15)/3! + (x²25)/5! - (x²35)/7! + ...

Now, we integrate each term of the power series with respect to x:

∫x²6×sin(x²5) dx = ∫(x²6 - (x²16)/3! + (x²26)/5! - (x²36)/7! + ...) dx

= (1/7)x²7 - (1/51)(x²17)/3! + (1/155)(x²27)/5! - (1/315)(x²37)/7! + ...

The power series representation of the given integral, centered at x = 0, is:

(x) = (1/7)x²7 - (1/51)(x²17)/3! + (1/155)(x²27)/5! - (1/315)(x²37)/7! + ...

The first 5 nonzero terms of the power series are:

(1/7)x²7 - (1/51)(x²17)/3! + (1/155)(x²27)/5! - (1/315)(x²37)/7! + (1/561)(x²47)/9! + ...

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The ordered pair of the equation is (1, - 1) or (3, 3).

The ordered pair of the equation is calculated by choosing a value of x and substituting it into the original equation and solving for the value of y as shown below.

The given equation is;

2x - y = 3

let x = 1

Now substitute the value of x into the original equation and solve for y as follows;

2x - y = 3

2 (1) - y = 3

2 - y = 3

y = 2 - 3

y = -1

We can also choose another value of x, say 3;

2(3) - y = 3

6 - y = 3

y = 3

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The value of k such that the vertical line x = k divides region R into two regions of equal area is approximately 0.7227.

What is Area?

Area is defined as the amount of two-dimensional space that a shape occupies. It can be calculated by multiplying two lengths of a shape, such as the length and width of a rectangle. Area units are always length units squared.

To find the area of region R, we need to integrate the function that defines the upper boundary of the region with respect to x. In this case, the upper boundary is given by y = 3sin(x/2).

To determine the area of R, we can integrate the function from x = 0 to x = (2pi)/3:

A = ∫[0, (2pi)/3] 3sin(x/2) dx

Using the integral property ∫sin(ax) dx = -1/a * cos(ax), we can rewrite the integral as:

A = -6 ∫[0, (2pi)/3] cos(x/2) dx

Evaluating the integral, we get:

A = -6 * [sin(x/2)]|[0, (2pi)/3]

Now we substitute the upper and lower limits into the equation:

A = -6 * [sin((2pi)/6) - sin(0)]

Since sin(0) = 0, the equation simplifies to:

A = -6 * sin((2pi)/6)

Simplifying further:

A = -6 * sin(pi/3)

Using the value of sin(pi/3) = sqrt(3)/2, we get:

A = -6 * (sqrt(3)/2)

A = -3sqrt(3)

However, area cannot be negative, so we take the absolute value:

|A| = 3sqrt(3)

Therefore, the area of region R is 3sqrt(3).

To find the value of k such that the vertical line x = k divides region R into two equal areas, we need to find the x-coordinate of the line of symmetry.

Let's assume the line of symmetry intersects x = k. We want the areas on both sides of the line to be equal, so the areas from x = 0 to x = k and from x = k to x = (2pi)/3 should be equal.

The total area of region R is 3sqrt(3), so the area on each side of the line of symmetry should be (1/2) * 3sqrt(3) = (3/2)sqrt(3).

Let's set up the equation to find k:

∫[0, k] 3sin(x/2) dx = (3/2)sqrt(3)

Using the same integral and evaluating it, we get:

-6 * [sin(x/2)]|[0, k] = (3/2)sqrt(3)

-6 * sin(k/2) = (3/2)sqrt(3)

Dividing both sides by -6 and multiplying by 2/3, we have:

sin(k/2) = -sqrt(3)/4

Taking the inverse sine (arcsin) of both sides, we get:

k/2 = arcsin(-sqrt(3)/4)

To find the value of k, we need to consider the range of the arcsine function. The range of arcsin(x) is -pi/2 ≤ arcsin(x) ≤ pi/2.

Since we're looking for the positive value of k, we need to consider the positive range of arcsin. Thus:

k/2 = arcsin(-sqrt(3)/4)

k = 2 * arcsin(-sqrt(3)/4)

Evaluating this expression using a calculator, we find:

k ≈ 0.7227

Therefore, the value of k such that the vertical line x = k divides region R into two regions of equal area is approximately 0.7227.

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The equation relating x and y when they vary inversely can be written as y = k/x. For each given set of values, substitute x = 5 into the equation to find the corresponding value of y.

To write an equation relating x and y when they vary inversely, we can use the formula y = k/x, where k is a constant.

Let's find the value of k for each given set of values:

a) x = 2, y = 9

Using the formula, we have 9 = k/2. Solving for k, we find k = 18. Therefore, the equation relating x and y is y = 18/x.

b) x = -1, y = 11

Using the formula, we have 11 = k/(-1). Solving for k, we find k = -11. Therefore, the equation relating x and y is y = -11/x.

c) x = 3, y = 27

Using the formula, we have 27 = k/3. Solving for k, we find k = 81. Therefore, the equation relating x and y is y = 81/x.

Now, let's find y when x = 5:

Therefore, when x = 5, the value of y varies depending on the given set of values. In case a), y = 3.6, in case b), y = -2.2, and in case c), y = 16.2.

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The given integral to be evaluated is,ZZZB (x2 + y2 + z2)1/2 dV,where B is the ball with center the origin and radius 2.So, using spherical coordinates, we getx = rsinθcosφy = rsinθsinφz = rcosθVolume element in spherical coordinates is given by dV = r2 sinθ dr dθ dφ

For the ball with center the origin and radius 2, the bounds are 0 ≤ r ≤ 2, 0 ≤ θ ≤ π, 0 ≤ φ ≤ 2π∴ The integral becomesZZZB (x2 + y2 + z2)1/2 dV = ∫02π∫0π∫20r3 sinθ dr dθ dφ= 2π ∫0π sinθ dθ ∫20 r3 dr= 2π ∫0π sinθ dθ [r4/4]20= 2π ∫0π sinθ dθ [16/4]= 2π[4]∫0π sinθ dθ= 8 π [–cosθ]0π= 16 π.Here, we used the spherical coordinates to evaluate the given integral. We first found the volume element in spherical coordinates and then used the given bounds for the ball with center the origin and radius 2.

In this problem, the given integral has to be evaluated. For this, we used the spherical coordinates. In spherical coordinates, we have three variables namely r, θ and φ. Here, r represents the distance of the point from the origin, θ represents the angle between the positive z-axis and the line segment joining the origin and the point, and φ represents the angle between the positive x-axis and the projection of the line segment joining the origin and the point onto the xy-plane.We used the spherical coordinates as we have a ball with center the origin and radius 2. The bounds of the ball are given by 0 ≤ r ≤ 2, 0 ≤ θ ≤ π, 0 ≤ φ ≤ 2π.Using these bounds, we evaluated the integral using the given formula for the volume element in spherical coordinates. On simplifying, we got the value of the integral as 16 π.

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The money Candy has at the start is £2.475.

We have,

The concept used in the given explanation is the concept of solving equations.

The explanation starts by assigning the variable M to represent the amount of money Candy had at the start.

The amounts spent on drinks and cake are given as £1.45 and £1.20, respectively. By adding these amounts together, the total amount spent is calculated as £1.65 (shown as (10)).

Now,

Let the money Candy had at the start = M

Now,

Amount spent:

Drinks = £1.45

Cake = £1.20

Total amount spent.

= 1.45 + 1.20

= £1.65 ______(10

Now,

Amount remaining.

= 2/3 M ______(2)

Now,

From (1) and (2),

We can make an equation as:

1.65 = 2/3 M

M = (1.65 x 3) / 2

M = £2.475

Thus,

The money Candy has at the start is £2.475.

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In an instruction like "z = x y," the symbols x, y, and z are examples of variables.

Variables are placeholders that represent unknown or changing values in mathematical expressions or equations. They allow us to generalize mathematical relationships and solve problems using algebraic methods.

In mathematics, variables are symbols that represent unknown or varying quantities. They are used to express mathematical relationships, equations, and formulas.

In the given instruction "z = x y," the symbols x, y, and z are variables.

In this case, x and y represent the input values, and z represents the output or result of the mathematical operation defined by the equation.

Variables are used extensively in algebra to solve equations, manipulate expressions, and analyze mathematical relationships. They enable us to express and solve problems symbolically, without knowing the specific values of the variables.

By assigning specific values to variables, we can evaluate expressions, solve equations, and find solutions to mathematical problems.

Variables can represent a wide range of quantities, including numbers, measurements, constants, or even abstract concepts. They provide flexibility and generality in mathematical modeling and problem-solving.

By using variables, we can establish connections between different mathematical quantities and derive meaningful conclusions based on their relationships.

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For x = 9 and y = 4, the value of

X square +2xy+y square -3 is

= 166

Given:-

[tex]\sf {\hookrightarrow \: {x}^{2} + 2xy + {y}^{2} - 3} \\ \\ [/tex]

[tex]\sf {\hookrightarrow {9}^{2} + 2(9)(4) + {4}^{2} - 3 } \\ \\ [/tex]

[tex]\sf {\hookrightarrow \: 81 + 72 + 16 - 3 } \\ \\[/tex]

[tex]\sf {\hookrightarrow 169 - 3 } \\ \\ [/tex]

[tex]\sf {\hookrightarrow \huge \boxed {166}} [/tex]

[tex] {\rule{200pt}{5pt}}[/tex]

____Hope you understood!____

Given statement solution is:- The function y = √(x + 3) - 2 with at least four integer points on the axes.

To plot points and graph the function y = √(x + 3) - 2, we can utilize technology tools such as graphing calculators or online graphing tools. I'll demonstrate how to use an online graphing tool called Desmos, which is user-friendly and widely accessible.

Please follow these steps to plot the points and graph the function:

Visit the Desmos website or use any other graphing tool of your choice.

Clear any existing equations or plots on the graph.

Enter the function y = sqrt(x + 3) - 2 into the equation field.

The graphing tool will automatically plot the function.

To plot integer points, simply choose four different integer values for x and calculate the corresponding y values.

Let's choose the following integer values for x: -3, -2, 0, and 2.

Calculate the corresponding y values:

For x = -3: y = sqrt(-3 + 3) - 2 = -2

For x = -2: y = sqrt(-2 + 3) - 2 = -1

For x = 0: y = sqrt(0 + 3) - 2 = 1

For x = 2: y = sqrt(2 + 3) - 2 = 0

Plot these points on the graph by clicking on the appropriate locations.

If you accidentally click on a point and want to delete it, simply click on the point again to remove it.

Make sure the graph displays the axes and all the plotted points.

By following these steps, you should be able to plot the points and graph the function y = √(x + 3) - 2 with at least four integer points on the axes.

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