in problems 11–16, find a general solution of the system x′1t2 = ax1t2 for the given matrix a.

We would need a sample size of approximately 598 men to have a 90% chance of detecting a true proportion of 8% with a significance level of 0.1.

To determine the sample size required, we need to perform a hypothesis test. The null hypothesis is that the proportion of men playing golf is 5%, and the alternative hypothesis is that it is greater than 5%.

We want to have a significance level (alpha) of 0.1, which corresponds to a confidence level of 0.9, and we also want a statistical power of 0.9. Assuming a one-tailed test, we can use a z-test to calculate the sample size needed.

Using a statistical calculator, we find that the critical value of z for a significance level of 0.1 is 1.28, and the critical value of z for a power of 0.9 is 1.28 + 1.28 = 2.56. The effect size is 0.03, which is the difference between the hypothesized proportion of 0.05 and the true proportion of 0.08. Plugging these values into the sample size formula for a z-test, we get:

n = ((1.28 + 2.56) / 0.03)² = 597.3

Therefore, we would need a sample size of approximately 598 men to have a 90% chance of detecting a true proportion of 8% with a significance level of 0.1.

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Function: A mathematical relationship that assigns each input value to a unique output value.

Combined Function: A function formed by applying one function to the output of another function.

Quadratic Function: A function with a polynomial equation of degree 2, represented as f(x) = ax² + bx + c, where a, b, and c are constants.

We have,

Function:

A function is a mathematical relationship or rule that assigns each input value (or element) from a set, called the domain, to a unique output value (or element) from another set, called the range.

Example:

Let's consider a function f(x) = 2x + 3.

This function takes an input value (x), multiplies it by 2, and then adds 3 to get the output value.

For example, if we input x = 4 into the function, we get f(4) = 2(4) + 3 = 11. So, the function maps the input value 4 to the output value 11.

Combined Function:

A combined function is formed by performing multiple operations on a given input value. It involves applying one function to the output of another function.

This allows us to express complex relationships between variables by combining simpler functions.

Example:

Let's consider two functions: f(x) = 2x and g(x) = x².

The combined function h(x) is formed by applying g(x) to the output of f(x). In other words, h(x) = g(f(x)).

If we input x = 3 into the combined function, we first evaluate f(x) = 2(3) = 6, and then evaluate g(6) = 6² = 36. So, h(3) = 36.

Quadratic Function:

A quadratic function is a type of function that can be represented by a polynomial equation of degree 2.

It has the general form f(x) = ax² + bx + c, where a, b, and c are constants.

The graph of a quadratic function is a parabola, which can open upwards or downwards depending on the value of the coefficient "a".

Example:

Let's consider the quadratic function f(x) = 2x² - 3x + 1.

This function has a coefficient of 2 for the x^2 term, -3 for the x term, and 1 for the constant term.

If we input x = 2 into the function,

We get f(2) = 2(2)² - 3(2) + 1 = 8 - 6 + 1 = 3.

So, the function maps the input value 2 to the output value 3.

The graph of this quadratic function is a parabola that opens upwards.

Thus,

Function: A mathematical relationship that assigns each input value to a unique output value.

Combined Function: A function formed by applying one function to the output of another function.

Quadratic Function: A function with a polynomial equation of degree 2, represented as f(x) = ax² + bx + c, where a, b, and c are constants.

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The properties of the functions are

A sinusoidal function is represented as

f(x) = Acos(2π/B(x + C)) + D or

f(x) = Asin(2π/B(x + C)) + D

Where the properties are

Using the above as a guide, we have the following:

4) y = sin 2(x - π/2):

5) y = cos 1/2(x - π):

6) y = 3cos 2(x + π) - 1:

The graphs of the functions are added as attachments

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The parametric equation for the ellipse with foci (0, −2), (8, −2), and vertex (9, −2) is ((x-9)^2/64) + (y+2)^2/36 = 1.

To find the equation for the ellipse with the given foci and vertex, we can use the standard form of the equation for an ellipse:

((x-h)^2/a^2) + ((y-k)^2/b^2) = 1,

where (h, k) is the center of the ellipse, a is the distance from the center to the vertex, and b is the distance from the center to the co-vertex. Since the foci are on the x-axis, the center of the ellipse is at (c, −2), where c is the distance from the center to a focus. Using the distance formula, we have:

c = √(8^2/4) = 4

The distance from the center to the vertex is a = 5, since the vertex is 5 units to the right of the center. The distance from the center to the co-vertex is b = 3, since the co-vertex is 3 units above or below the center. Substituting these values into the standard form of the equation, we get:

((x-9)^2/25) + (y+2)^2/9 = 1

Since the foci are on the x-axis, we have:

2c = 8, or c = 4

The distance from the center to the vertex is a = 5, so:

a^2 = 25

Using the relationship between a, b, and c for an ellipse, we have:

b^2 = a^2 - c^2 = 25 - 16 = 9

Substituting these values into the standard form of the equation, we get:

((x-9)^2/64) + (y+2)^2/36 = 1

Therefore, the equation for the ellipse with foci (0, −2), (8, −2), and vertex (9, −2) is ((x-9)^2/64) + (y+2)^2/36 = 1.

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Rounded to the nearest ten-thousandth, the probability is 0.9375.

Probability is a branch of mathematics in which the chances of experiments occurring are calculated. It is by means of a probability, for example, that we can know from the chance of getting heads or tails in the launch of a coin to the chance of error in research.

Let's assume that the probability of a randomly selected motorist driving more than 5 miles per hour over the speed limit is p. Then, the probability of a motorist not driving more than 5 miles per hour over the speed limit is 1-p.

The probability of at least one motorist driving more than 5 miles per hour over the speed limit can be found by using the complement rule. That is:

P(at least one motorist driving more than 5 miles per hour over the speed limit) = 1 - P(no motorist driving more than 5 miles per hour over the speed limit)

The probability of no motorist driving more than 5 miles per hour over the speed limit can be found by using the binomial distribution. Since there are 4 motorists and each one has a probability of 1-p of not driving more than 5 miles per hour over the speed limit, the probability is:

P(no motorist driving more than 5 miles per hour over the speed limit) = (1-p)⁴

Therefore, the probability of at least one motorist driving more than 5 miles per hour over the speed limit is:

P(at least one motorist driving more than 5 miles per hour over the speed limit) = 1 - (1-p)⁴

We are not given a specific value for p, so we cannot calculate the probability exactly. However, if we assume that p = 0.5 (i.e., there is a 50-50 chance of a randomly selected motorist driving more than 5 miles per hour over the speed limit), then the probability of at least one motorist driving more than 5 miles per hour over the speed limit is:

P(at least one motorist driving more than 5 miles per hour over the speed limit) = 1 - (1-0.5)⁴ = 0.9375

Rounded to the nearest ten-thousandth, the probability is 0.9375. However, if we assume a different value for p, the probability will be different.

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Answer:

LJ = 13.12

Step-by-step explanation:

given the triangles are similar then the ratios of corresponding sides are in proportion, that is

[tex]\frac{LJ}{ZX}[/tex] = [tex]\frac{JK}{XY}[/tex] ( substitute values )

[tex]\frac{LJ}{8.2}[/tex] = [tex]\frac{13.92}{8.7}[/tex] ( cross- multiply )

8.7 × LJ = 8.2 × 13.92 = 114.144 ( divide both sides by 8.7 )

LJ = [tex]\frac{114.144}{8.7}[/tex] = 13.12

Factored 25x^2 - 36 as the product of (5x + 6) and (5x - 6). To factor completely 25x^2 - 36, we first note that both 25 and 36 are perfect squares. Specifically, 25 = 5^2 and 36 = 6^2.

Using the difference of squares identity, we can write:

25x^2 - 36 = (5x)^2 - 6^2

Now, we can use the difference of squares formula again to obtain:

25x^2 - 36 = (5x + 6)(5x - 6)

In general, when factoring a quadratic expression of the form ax^2 + bx + c, where a, b, and c are constants, it is helpful to look for common factors or perfect squares first. The difference of squares formula can also be a useful tool in factoring quadratic expressions.

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The height of the tree is 6.08 meters.

We are given that;

Base to kays feet= 4.75m, kays feet to end of shadow=1.25m, kays height=1.60m

Now,

To find the height of the tree, you need to use similar triangles. The ratio of the corresponding sides of similar triangles is equal, so you can set up a proportion between the heights and the shadow lengths. You can write your solution as:

1.60/1.25 = h/4.75 h = 1.60/1.25 x 4.75 h = 6.08

Therefore, by the proportions the answer will be 6.08 meters.

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[tex]\begin{array}{llll} \textit{using the pythagorean theorem} \\\\ a^2+o^2=c^2\implies a=\sqrt{c^2 - o^2} \end{array} \qquad \begin{cases} c=\stackrel{hypotenuse}{37}\\ a=\stackrel{adjacent}{x}\\ o=\stackrel{opposite}{35} \end{cases} \\\\\\ x=\sqrt{ 37^2 - 35^2}\implies x=\sqrt{ 1369 - 1225 } \implies x=\sqrt{ 144 }\implies x=12[/tex]

We can use the power series representation sum of the function f(x) = (1+x)^3 to find a closed-form expression for the series x[infinity] n=1 (-1)^n (2n+1)^3n.

Specifically, we have:

f(x) = (1+x)^3 = 1 + 3x + 3x^2 + x^3

Taking the cube of this expression gives:

f(x)^3 = (1 + 3x + 3x^2 + x^3)^3

Expanding this out using the binomial theorem gives:

f(x)^3 = 1 + 9x + 36x^2 + 84x^3 + 126x^4 + 126x^5 + 84x^6 + 36x^7 + 9x^8 + x^9

We can rewrite the terms with even powers of x as:

f(x)^3 = 1 + 9x + 36x^2 + 84x^3 + x^4 (126 + 126x + 84x^2 + 36x^3 + 9x^4)

Note that the expression in parentheses is just the power series representation of (1+x)^4. Therefore, we can simplify the above expression to:

f(x)^3 = 1 + 9x + 36x^2 + 84x^3 + x^4 (1+x)^4

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The true percentage of Florida teenagers who value helping others in need lies between 76.1% and 81.9%.we can estimate with 95% confidence that the number of Florida teenagers who value helping others in need is between interval 1,596,900 and 1,722,900.

According to a survey of 800 Florida teenagers, 79% of them said that helping others in need will be very important to them as adults.

However, due to the limitations of a sample survey, this percentage might not be an exact representation of the entire population of Florida teenagers.

To estimate the true percentage of Florida teenagers who value helping others in need, a confidence interval can be used.

The margin of error given in the survey is +/- 2.9%, which means that we can be confident that the true percentage lies within a range of 2.9% above or below the sample percentage of 79%.

To calculate the confidence interval, we need to find the upper and lower bounds of the range. To find the lower bound, we subtract the margin of error from the sample percentage:

Lower bound = 79% - 2.9% = 76.1%

To find the upper bound, we add the margin of error to the sample percentage:

Upper bound = 79% + 2.9% = 81.9%

Therefore, we can say with 95% confidence that the true percentage of Florida teenagers who value helping others in need lies between 76.1% and 81.9%.

If we assume that the population of Florida teenagers is 2.1 million, we can also estimate the range of the number of teenagers who value helping others in need. To do this, we multiply the lower and upper bounds of the confidence interval by the population size:

Lower bound = 76.1% x 2.1 million = 1,596,900 teenagers

Upper bound = 81.9% x 2.1 million = 1,722,900 teenagers

Therefore, we can estimate with 95% confidence that the number of Florida teenagers who value helping others in need is between 1,596,900 and 1,722,900.

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a) The firm’s long-run total, average, and marginal cost functions is:

C = wl + vk = (0.1w + 0.2v)q

AC = C/q = 0.1w + 0.2v

MC = dC/dq = 0.1w + 0.2v

b) The firm’s short-run total, average, and marginal cost functions:

C = wl + 10v = 0.1wq + 10v

AC = C/q = 0.1w + 10v/q

MC = dC/dq = 0.1w

c) This firm’s long-run and short-run average and marginal cost curves.

Long run cost:

AC = 0.3 + 0.2 = 0.5

MC = 0.3 + 0.2 = 0.5

Short run:

AC = 0.3 + 10/q

MC = 0.3

Cost function shows the relationship between the cost of production and the level of output. In the short run a portion of the total cost is fixed but, in the long run, all cost are variable. Average cost equals the cost per unit (i.e., total cost divided by output) and the marginal cost equals the change in cost per unit change in output.

The producer used k and l such that 5k = 10l

The output q, then, is

q = 5k = 10l

i.e., k = 1/5q = 0.2q and l = 1/10q = 0.1q

Long run cost:

C = wl + vk = (0.1w + 0.2v)q

AC = C/q = 0.1w + 0.2v

MC = dC/dq = 0.1w + 0.2v

b) Suppose that k is fixed at 10 in the short run. Calculate the firm's short-run total, average and marginal cost functions.

b) K= 10,

Short run cost:

C = wl + 10v = 0.1wq + 10v

AC = C/q = 0.1w + 10v/q

MC = dC/dq = 0.1w

c) Long run cost:

AC = 0.3 + 0.2 = 0.5

MC = 0.3 + 0.2 = 0.5

Short run:

AC = 0.3 + 10/q

MC = 0.3

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If a = 4, then a 2 · a 3 is equivalent to all of the following except _ 4^2 · 4^3 = 1,024

Noted that Expression in maths is defined as the collection of numbers variables and functions by using signs like addition, subtraction, multiplication, and division.

We are given that a = 4, then the expression could be;

a^2 · a ^3

Substitute the values;

a^2 · a ^3  = 4^2 · 4^3

= 16 . 64

= 1,024

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The statement is true. With a classic update using linear function approximation, we will always converge to some values, but they may not be optimal. This is because linear function approximation only allows for a limited representation of the value function, and the approximated function may not capture the true underlying structure of the problem.

Linear function approximation is commonly used in reinforcement learning to estimate the value function. The idea is to approximate the value function using a linear combination of features. During the learning process, the weights of the linear combination are updated using the classic update rule. While this approach is computationally efficient, it can result in suboptimal policies. The reason for this is that the approximated function may not be able to capture the complexity of the problem. This can lead to inaccuracies in the value function estimates, which in turn can result in suboptimal policies. To address this issue, more advanced function approximation methods, such as neural networks, can be used to approximate the value function. These methods can capture more complex relationships in the data and provide more accurate estimates of the value function.

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the lateral area of the figure to the right is 160 m². Option C

From the information given, the shape is a triangular prism.

Thus, the formula for calculating the lateral area of a triangular prism is expressed as;

A = (a + b + c)h

Such that the parameters of the formula are;

Substitute the values, we have;

Lateral area = 16(10)

Multiply the values

Lateral area = 180 m²

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Answer:

a) 1/6

b)1/2

c)5/6

d)1/6

Step-by-step explanation:

because the probability of rolling one number is one in six we can add however many other probabilities

There is not enough evidence to conclude that the average time spent on leisure activities by American adults has changed. Sample standard deviation (s) is 8.984 hours. Sample mean (x') is 19.9 hours per week

To test whether the claim that American adults spend an average of 18 hours per week on leisure activities is true or false, we can conduct a hypothesis test.

First, we need to define the null and alternative hypotheses. Let µ be the population mean time spent on leisure activities by American adults.

Null hypothesis: µ = 18 hours per week

Alternative hypothesis: µ ≠ 18 hours per week

We can then calculate the sample mean and standard deviation from the data given as follows:

Sample mean (x') = (13+4+15+9+21+42+15+34+20+6) / 10 = 19.9 hours per week

Sample standard deviation (s) = 8.984 hours

Next, we can calculate the test statistic (t-value) using the formula:

t = (x' - µ) / (s / √(n))

where n is the sample size (10).

Using a t-distribution with 9 degrees of freedom (n-1), we can find the critical t-value at a 5% significance level to be ±2.306.

We calculate the t-value as:

t = (19.9 - 18) / (8.984 / √(10)) = 0.911

Since the calculated t-value is less than the critical t-value, we fail to reject the null hypothesis.

In other words, the claim that American adults spend an average of 18 hours per week on leisure activities is not contradicted by the sample data.

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The correct answer is D, open market purchase, increasing the reserve requirement, and decreasing the discount rate.

When the Fed judges inflation to be the most significant problem in the economy and wishes to employ all three of its policy instruments, it will implement expansionary monetary policy.

This involves increasing the money supply in the economy to stimulate growth and reduce inflation.
To do this, the Fed will conduct open market operations, which involve purchasing government securities from banks.

This injects money into the banking system and increases the amount of reserves banks have available to lend out. This increase in lending stimulates economic growth and reduces inflation.
In addition to open market operations, the Fed will increase the reserve requirement, which is the amount of money that banks are required to hold in reserve.

This reduces the amount of money banks have available to lend out and helps to control inflation.
Finally, the Fed will decrease the discount rate, which is the interest rate at which banks can borrow money from the Fed.

This makes it cheaper for banks to borrow money and encourages them to lend more, stimulating growth and reducing inflation.
In summary, when the Fed judges inflation to be the most significant problem in the economy and wishes to employ all three of its policy instruments, it will implement expansionary monetary policy by conducting open market operations, increasing the reserve requirement, and decreasing the discount rate.

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Answer:

Step-by-step explanation:

The work done by f in moving a particle once counterclockwise around the given curve is -15π.

The problem requires us to calculate the work done by the vector field f along a closed curve C, which is a circle centered at (5,5) with a radius of 5. To do this, we can use the line integral of f along C, which is given by:

∫C f · dr = ∫C (f(x,y) · T) ds

where T is the unit tangent vector to C and ds is the arc length element along C.

To parameterize the curve C, we can use the parametric equations:

x = 5 + 5cos(t)

y = 5 + 5sin(t)

with 0 ≤ t ≤ 2π. Then, the unit tangent vector T is given by:

T = (-sin(t), cos(t))

and the arc length element ds is given by:

ds = √(x'(t)² + y'(t)²) dt = 5 dt

Using these expressions, we can compute the line integral as:

∫C f · dr = ∫C [(x-3y)i + (3x-y)j] · (-sin(t)i + cos(t)j) 5 dt

After some algebraic manipulation, we obtain:

∫C f · dr = -15π

Therefore, the total work done by f in moving the particle once around the given curve counterclockwise is -15π.

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The work done by f in moving a particle once counterclockwise around the given curve is zero.

To find the work done by a vector field f in moving a particle along a curve C, we use the line integral formula. The line integral of a vector field f along a curve C is given by the formula ∫C f · dr, where dr is the differential of the position vector r(t) of the curve C. In this case, the vector field is f = (x - 3y)i + (3x - y)j and the curve is the circle (x - 5)² + (y - 5)² = 25 centered at (5,5) with radius 5. To evaluate the line integral, we need to parameterize the curve. Since the curve is a circle, we can use the parametrization r(t) = 5cos(t)i + 5sin(t)j, where t ranges from 0 to 2π. Then, dr = -5sin(t)dt i + 5cos(t)dt j.

Evaluating the line integral, we get ∫C f · dr = ∫0^2π f(r(t)) · dr/dt dt = ∫0^2π (-15sin²(t) + 15cos²(t))dt = 0. Therefore, the work done by f in moving a particle once counterclockwise around the given curve is zero.

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The  solution to the system of differential equations with initial conditions x(0) = -33 and y(0) = 12 is:

x(t) = 15e^(11t), y(t) = -24e^(-2t)

To solve the system of differential equations {x' = 11x + 24y, y' = -3x - 6y}, we can use the method of matrix exponentials. First, we write the system in matrix form:

{{x'}, {y'}} = {{11, 24}, {-3, -6}} {{x}, {y}}

Next, we compute the matrix exponential of the coefficient matrix:

e^(tA) = {{e^(11t), 4e^(11t)}, {-3e^(-2t), e^(-2t)}}

Then, we can use this matrix exponential to find the solution to the system of differential equations:

{{x(t)}, {y(t)}} = e^(tA) {{x(0)}, {y(0)}}

Plugging in the initial conditions x(0) = -33 and y(0) = 12, we get:

{{x(t)}, {y(t)}} = {{-33e^(11t) + 4(12)e^(11t)}, {-3(12)e^(-2t) + 12e^(-2t)}}

Simplifying, we get:

x(t) = -33e^(11t) + 48e^(11t) = 15e^(11t)

y(t) = -36e^(-2t) + 12e^(-2t) = -24e^(-2t)

Therefore, the solution to the system of differential equations with initial conditions x(0) = -33 and y(0) = 12 is:

x(t) = 15e^(11t), y(t) = -24e^(-2t)

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Therefore, The exponential function approaches zero as the input approaches negative infinity, and as x increases towards infinity, the value of e−3x approaches zero.

(a) The right end behavior of y = log6(x) as x approaches infinity is that y approaches negative infinity. This is because as x increases towards infinity, the value of log6(x) becomes larger and larger negative values. Explanation: The logarithm function approaches negative infinity as the input approaches zero, and as x increases towards infinity, the value of log6(x) approaches negative infinity.
(b) The right end behavior of y = e−3x as x approaches infinity is that y approaches 0. This is because as x increases towards infinity, the exponent -3x becomes larger and larger negative values, making the value of e−3x approach zero.  

Therefore, The exponential function approaches zero as the input approaches negative infinity, and as x increases towards infinity, the value of e−3x approaches zero.

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In a single elimination tournament, a team plays until it loses. Therefore, for a tournament with 8 teams, 7 games must be played. The number of games in a single elimination tournament equals the number of teams minus 1.

Among the random variables you mentioned - z, t, chi-square, and F - the ones that have only nonnegative values are the chi-square and F distributions.

The chi-square (χ²) distribution is a special case of the gamma distribution, and it is used extensively in hypothesis testing and statistical modelling.

It is defined for nonnegative values, as it represents the sum of squared independent standard normal random variables.
The F-distribution, named after statistician Sir Ronald A. Fisher, is another continuous probability distribution that is defined only for nonnegative values. It is commonly used in the analysis of variance (ANOVA) to test the equality of multiple group means or in regression analysis to test the overall significance of a model.
In contrast, both the z (standard normal) and t (Student's t) distributions are defined for values across the entire real number line, including positive, negative, and zero values. The z-distribution is used for hypothesis testing and confidence intervals in situations where the population standard deviation is known, while the t-distribution is used when the population standard deviation is unknown and estimated from the sample data.

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When a tax is imposed, some of the lost surplus is converted to tax revenue, while the rest is deadweight loss.

A tax creates a wedge between the price paid by buyers and the price received by sellers, reducing the quantity of goods traded in the market. This reduction in quantity causes a loss in surplus, which is the sum of consumer surplus and producer surplus. However, some of this lost surplus is converted to tax revenue, which is the amount of money collected by the government from the tax. The amount of lost surplus converted to tax revenue depends on the price elasticity of demand and supply in the market. If the demand and supply are relatively inelastic, a larger share of the lost surplus is converted to tax revenue. On the other hand, if the demand and supply are relatively elastic, a smaller share of the lost surplus is converted to tax revenue. The rest of the lost surplus that is not converted to tax revenue is called deadweight loss, which represents the reduction in economic welfare that is not compensated by the tax revenue. Deadweight loss occurs because the tax creates a distortion in the market that reduces the efficient allocation of resources, leading to inefficiencies in production and consumption.

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c = 0.71, which means that the probability of obtaining a z-score greater than 0.71 in a standard normal distribution is 0.2445. We need to use a standard normal distribution table or calculator.

From the given information, we know that the area to the right of z (which is c in this case) is 0.2445. Looking up this value in a standard normal distribution table, we find that the z-score that corresponds to this area is approximately 0.71. Therefore, c = 0.71. We can also use a calculator to find the value of c. Using the inverse normal function (also known as the z-score function) on a calculator or spreadsheet, we can input the area to the right of c (0.2445) and get the corresponding z-score, which is 0.71.

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