Ms. Avila got blue ribbons to give to participants at the school science fair. She ordered 120
blue ribbons from the Smart Start school supplier. Since this was a bulk order, Smart Start
reduced the price of each blue ribbon by $0.75. The total came to $60.
Which equation can you use to find the amount of money, b, Smart Start normally charges
for a blue ribbon?

The solution to the system of equations is (0.839, 1.129).

What is a graph ?

Graph can be defined as visual representation of the data through the given ordered pairs .

Given ,

The graph of a line passes through the points (0, 0.5) and (6, 5). A second i has the equation 5y = 7-6x.

The equation of the first line can be found using the two given points, using the slope-intercept form:

y = mx + b

where m is the slope and b is the y-intercept. To find the slope, we can use the formula:

m = (y2 - y1) / (x2 - x1)

Plugging in the values from the two points, we have:

m = (5 - 0.5) / (6 - 0) = 4.5 / 6 = 0.75

So the equation of the first line is:

y = 0.75x + 0.5

To find the solution to the system of equations, we need to find the point where the two lines intersect, which means finding the values of x and y that satisfy both equations. To do this, we can substitute the equation of the first line into the second equation:

5y = 7 - 6x

0.75x + 0.5 = 7 - 6x

Solving for x, we get:

7.75x = 6.5

x = 6.5 / 7.75 = 0.839

Substituting x back into either equation to find y, we get:

y = 0.75x + 0.5 = 0.75 * 0.839 + 0.5 = 0.629 + 0.5 = 1.129

So the solution to the system of equations is (0.839, 1.129).

Therefore, The solution to the system of equations is (0.839, 1.129).

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The volume V of the solid obtained by rotating the region bounded by the given curves y = 2[tex]e^{-x}[/tex], y = 2, x = 4; about y = 4 is 256π.

The volume of the solid obtained by rotating the region bounded by the given curves about the line y=4 is given by

V = 2π ∫⁴₀ (4-2[tex]e^{-x}[/tex])² dx

We can calculate the integral using integration by parts.

Let u = 4-2[tex]e^{-x}[/tex]

⇒ du = 2d[tex]e^{-x}[/tex]

and dv = 2dx

⇒ v = x

So,

V = 2π ∫⁴₀ (4-2[tex]e^{-x}[/tex])² dx

 = 2π (x (4-2[tex]e^{-x}[/tex])²  - 0 (4-2[tex]e^{-x}[/tex]})²  - 2∫⁴₀ x d (4-2[tex]e^{-x}[/tex]))

 = 2π (2×4³ - 4∫⁴₀ x [tex]e^{-x}[/tex] dx)

Using integration by parts again

Let u = 4-2[tex]e^{-x}[/tex] ⇒ du = -2d[tex]e^{-x}[/tex]

and dv = xdx ⇒ v = x²/2

So,

V = 2π (2×4³ - 4∫⁴₀ x [tex]e^{-x}[/tex] dx)

 = 2π (2×4³ - 2×2∫⁴₀ x² [tex]e^{-x}[/tex] dx)

Again we can calculate the integral using integration by parts.

Let u = x² ⇒ du = 2xdx

and dv = [tex]e^{-x}[/tex] dx ⇒ v = -[tex]e^{-x}[/tex]

So,

V = 2π (2×4³ - 2×2∫⁴₀4 x^2 [tex]e^{-x}[/tex]} dx)

 = 2π (2 × 4³-2 × 2 × ([tex]-e^{-4}[/tex] + 2∫⁴₀  [tex]e^{-x}[/tex]dx))

 = 2π (2×4³ - 2×2 (1+4))

 = 2π (2×4³ - 16)

 = 2π (128-16)

 = 256π

Therefore, the volume V of the solid obtained by rotating the region bounded by the given curves y = 2[tex]e^{-x}[/tex], y = 2, x = 4; about y = 4 is 256π.

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The friend bought 4.725 pounds of grape, which is 3/8 of the amount that Martin bought

An equation is an expression that shows the relationship between two or more numbers and variables. An equation can either be linear, quadratic, cubic and so on

Mixed fraction is a fraction that is made up of a whole number and a fraction combined together.

Martin bought 12 3/5 pounds of grapes for a picnic.

12 3/5 pound is a mixed fraction

12 3/5 pound = 63/5 pounds

His friend bought 3/8 of that amount. Hence:

Amount of grapes the friend bought = (3/8) * 63/5 pounds = 4.725 pounds

The friend bought 4.725 pounds of grape

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Note that the missing values in the table are:

1) A - 28 : 20

2) B: - 63 : 49.

Note that this is a test of ratios and proportionality.

Note that the first tape comprises 7 boxes for the Fruit Mass and

the second tape 5 boxes for the Buiscuit mass.

So we know that the the figures represent Lowest Common Multiples when we are presented with the table.

From the table, we see that Fruitmass has a related number which is 28. Immediately we can see the relationship.

7 x 4 = 28

This means that the missing number can be arrived at by saying:

5 x 4 = 20

Repeat this same logic for the missing number in B to obtain: 63: 49.

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The covariance between the number of coins and the value of the coins is 240.

The covariance between the number of coins (N) and the value of the coins (V) can be calculated as

Cov(N, V) = Cov(n nickels + n dimes, 5n nickels + 10n dimes)

= 5 Cov(n nickels, n nickels) + 10 Cov(n nickels, n dimes) + 5 Cov(n dimes, n nickels) + 10 Cov(n dimes, n dimes)

= 5 * Var(n nickels) + 10 * Cov(n nickels, n dimes) + 5 * Cov(n dimes, n nickels) + 10 * Var(n dimes)

= 5 * 8 + 10 * 5 + 5 * 5 + 10 * 10

= 40 + 50 + 50 + 100

= 240

So the covariance between the number of coins and the value of the coins is 240.

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The Pythagorean Theorem shows that the triangle is acute

From the question, we have the following parameters that can be used in our computation:

Using the Pythagorean Theorem, we have

87^2 < 59^2 + 79^2

Evaluate

7569 < 9722

Since 7569 is less than 9722, we can conclude that the triangle is an acute triangle, not a right or obtuse triangle.

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The bond yield that comes closest to 7.75%, the  amount yield to maturity on an annual-pay bond, is 7.61%.

A. The bond yield that comes closest to 7.75%, the yield to maturity on an annual-pay bond, is 7.61%. A measure of return that considers the market parameters used to determine the return from an investment in a bond is called the bond-equivalent yield. It accounts for the time value of money, the coupon rate, and the current market rate to determine the yearly rate of return that would be generated if the bond were kept for a year. When comparing bonds with differing coupon rates and maturities, the bond-equivalent yield is utilised. It is a helpful tool for investors who want to compare various bonds and decide on their investments with knowledge.

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Sofia has 7 oranges

Ratio is a term that is used to compare two or more numbers. It is used to indicate how big or small a quantity is when compared to another.

The ratio of 700:500 can be written as 700/500

The ratio can be reduced to the lowest terms for easy calculation

We have 7/5

Let x represent orange

7/5 = x/5

By cross multiplying

35 =: 5x

Divide both sides of the equation by 5

35/5 = 5x/5

X = 7

Hence, if she has 5 watermelons, she will has 7 oranges

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The complete answers regarding flow of people and occupancy are solved as follows:

(a) The flow of people from downtown to campus is the rate at which the shuttle buses can transport passengers from the downtown bus stop to the campus bus stop.

(b) The flow of people from campus to downtown is equal to the rate at which passengers are coming to the campus bus stop.

(c) The average bus occupancy from campus to downtown is equal to the number of passengers on each bus divided by the number of seats on each bus.

Dispatching the buses from downtown to campus at equal headways would reduce the average waiting time of passengers going from campus to downtown. This is because the slower bus would have more time to pick up and drop off passengers, and the faster bus would not be slowed down by having to wait for the slower bus. The overall result would be a reduction in the average waiting time for all passengers.

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Complete Question:

Suppose that two shuttle buses operate (without coordination) between the campus and downtown. One bus can carry 20 people and has a round trip time of 8 min (including loading and unloading passengers), the other can carry 25 people but has a round trip time of 10 min. There is always a queue of people waiting at the downtown bus stop, but passengers come to the campus bus stop at a steady rate of 2/min. If neither bus is idle at any time, determine:

(a) The flow of people from downtown to campus.

(b) The flow of people from campus to downtown.

(c) The average bus occupancy from campus to downtown.

If one insisted on dispatching these buses from downtown to campus at equal headways, we would only slow down the faster bus. Would you expect this strategy to reduce the average waiting time of passengers going from campus to downtown?

a) The probability that the 3 balls chosen were all green is 1/21.

b) The probability that all the balls are chosen is red is 1/252.

What is probability?

Probability is a mathematical concept that measures the likelihood or chance of an event occurring. It is expressed as a number between 0 and 1, where 0 represents an impossible event and 1 represents a certain event.

(a) Given 3 balls were chosen at random without replacement, the probability that they were all green is given by:

P(green) = (number of favorable outcomes) / (number of total outcomes)

The number of favorable outcomes is the number of ways to choose 3 green balls out of 4, which is C(4,3) = 4.

The number of total outcomes is the number of ways to choose 3 balls out of 9, which is C(9,3) = 84.

So, the probability that 3 balls chosen were all green is:

P(green) = C(4,3) / C(9,3) = 4 / 84 = 1/21

(b) The probability that all the balls chosen are red is given by:

P(red) = (number of favorable outcomes) / (number of total outcomes)

The number of favorable outcomes is the number of ways to choose 5 red balls out of 5, which is C(5,5) = 1.

The number of total outcomes is the number of ways to choose 5 balls out of 9, which is C(9,5) = 126.

So, the probability that all the balls chosen are red is:

P(red) = C(5,5) / C(9,5) = 1 / 126 = 1/252

Hence,

a) The probability that the 3 balls chosen were all green is 1/21.

b) The probability that all the balls are chosen is red is 1/252.

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The triangle with the greater perimeter has angles 55, 15, and 112 degrees.

In a triangle, the sum of the interior angles is 180 degrees. Thus, if we have two non-congruent triangles with one angle equal (in this case, B = 55 degrees), we can set up an equation to solve for the third angle of the triangle with the greater perimeter.

Let's call the third angle in the first triangle x. Then, x + 55 + b = 180 degrees, or x + b = 125 degrees.

Since we know that a + b + B = 180 degrees for any triangle, we can set up a similar equation for the second triangle: y + 55 + a = 180 degrees, or y + a = 125 degrees.

Since the perimeter of a triangle is equal to the sum of its side lengths, and we know that a = 15 and b = 13, we can set up another equation to find the triangle with the greater perimeter:

Perimeter 1 = 15 + 13 + x and Perimeter 2 = 15 + 13 + y.

Solving for x, we find that x = 125 - b = 125 - 13 = 112 degrees. Solving for y, we find that y = 125 - a = 125 - 15 = 110 degrees.

Since 112 > 110, the triangle with the greater perimeter has angles 55, 15, and 112 degrees.

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The triangle with the greater perimeter is the one with the greater value of c, which we can calculate using the law of cosines and the law of sines.

Given two noncongruent triangles with angle B equal to 55 degrees and sides a and b, we need to find the measures of the angles of the triangle with the greater perimeter.

First, let's use the law of cosines to find the measure of angle A in each triangle:

[tex]c^2 = a^2 + b^2 - 2ab * cos(A)\\\\c = \sqrt{(a^2 + b^2 - 2ab * cos(A))} \\\\cos(A) = (a^2 + b^2 - c^2) / 2ab[/tex]

For the first triangle, we can use the given values:

[tex]cos(A) = (15^2 + 13^2 - c^2) / 2 * 15 * 13\\\\c = \sqrt{(15^2 + 13^2 - 2 * 15 * 13 * cos(A))}[/tex]

For the second triangle, we can use a and b as the lengths of the sides opposite angles A and C, respectively. Then we can use the law of sines to find the lengths of the other sides:

[tex]sin(A) / a = sin(C) / c\\\\c = a * sin(C) / sin(A)[/tex]

Substituting values and solving, we get:

[tex]c = 15 * sin(C) / sin(A)\\\\cos(A) = (15^2 + b^2 - c^2) / 2 * 15 * b\\\\c = \sqrt{(15^2 + b^2 - 2 * 15 * b * cos(A))}[/tex]

Now we can compare the two perimeters (the sum of the lengths of all sides) to determine which triangle has the greater perimeter. The perimeter of each triangle will be a + b + c.

Whichever triangle has the larger value of c will have the greater perimeter, as a and b are the same in both triangles.

So, we can use the value of c that we just calculated for each triangle to determine which triangle has the greater perimeter, and therefore, which triangle has the greater measure of its angles.

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The integral which represents the area between f(x) = x^4 is [tex]A = \int\limits^2_0 {x^4} \, dx[/tex] and the solution is A = 32/5.

A theorem that connects the ideas of integrating and differentiating functions is known as the fundamental theorem of calculus. By calculating the difference between the antiderivative at the higher and lower limits of the integration process, the fundamental theorem of calculus supports the method.

The given function of the area is:

f(x) = x^4.

The integral which represents this area can be written as:

[tex]A = \int\limits^2_0 {x^4} \, dx[/tex]

Using the fundamental Theorem of Calculus we have:

[tex]A = \int\limits^2_0 {x^4} \, dx\\\\A = [\frac{x^5}{5} ]_0^2[/tex]

Substituting the value of limits we have:

A = [2^5 / 5] = 32/5

Hence, the integral which represents the area between f(x) = x^4 is [tex]A = \int\limits^2_0 {x^4} \, dx[/tex] and the solution is A = 32/5.

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The composite function f (ƒ-¹ (5)) is 5

From the question, we have the following parameters that can be used in our computation:

f(x) = (x + 10)/5

Also, we have

f-¹ (x) = 5x - 10

The composite function f (ƒ-¹ (5)) is calculated as

f (ƒ-¹ (x)) = (5x - 10 + 10)/5

Evaluate the difference

f (ƒ-¹ (x)) = 5x/5

So, we have

f (ƒ-¹ (x)) = x

Substitute 5 for x

f (ƒ-¹ (5)) = 5

Hence, the solution is 5

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Complete question

Use the given function to find the following.

f(x) = (x + 10)/5

f-¹ (x) = 5x - 10

Find f (ƒ-¹ (5)). (no spaces)

The length of the ramp is solved to be 15.33 feet

The length of the ramp is worked using SOH CAH TOA

Sin = opposite / hypotenuse - SOH

Cos = adjacent / hypotenuse - CAH

Tan = opposite / adjacent - TOA

The direction of movements describes a right triangle of

adjacent = 15 feet

hypotenuse = length of the ramp = ?

The length of the ramp is calculated using cos, CAH

let the angle be x

cos x = adjacent / hypotenuse

cos 12 =  15 / hypotenuse

hypotenuse = 15 / cos 12

hypotenuse = 15.334

length of ramp = 15.33 feet

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The general solution to the differential equation  [tex]3x^2y' + 6xy = 15y^3[/tex]  is not possible to obtain in a closed form as it is a nonlinear differential equation.

One way to find a numerical solution to this equation is to use numerical methods such as the Runge-Kutta method, the Euler method, or finite difference methods. Another way is to use numerical software, such as MATLAB, to obtain an approximate solution. To find an analytical solution, one can try to transform the equation into a more manageable form using substitution or other techniques, but it may not be possible to obtain an exact solution. The general solution to the differential equation [tex]3x^2y' + 6xy = 15y^3[/tex]  is not possible to obtain in a closed form as it is a nonlinear differential equation.

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It is not possible to guarantee the exact number of points within a certain distance of each other in a square with 42 points placed inside.

Since the side length of the square is 2, the area of the square is 2 x 2 = 4. If 42 points are placed inside the square, the maximum distance between any two points will be equal to the diagonal of the square, which is equal to the square root of 2 times the side length of the square. The diagonal of the square is 2 * √(2), or approximately 2.82.

The number of points within a certain distance of each other depends on the size of the distance, as well as the density of the points in the square. However, it is not possible to guarantee that all 42 points will be within a certain distance of each other. The best we can say is that most of the points will be relatively close to each other, but some may be farther apart. The exact number of points that are within a certain distance of each other will depend on the specific arrangement of the points.

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It is not possible to guarantee the exact number of points within a certain distance of each other in a square with 42 points placed inside.

Since the side length of the square is 2, the area of the square is 2 x 2 = 4. If 42 points are placed inside the square, the maximum distance between any two points will be equal to the diagonal of the square, which is equal to the square root of 2 times the side length of the square. The diagonal of the square is 2 * √(2), or approximately 2.82.

The number of points within a certain distance of each other depends on the size of the distance, as well as the density of the points in the square. However, it is not possible to guarantee that all 42 points will be within a certain distance of each other. The best we can say is that most of the points will be relatively close to each other, but some may be farther apart. The exact number of points that are within a certain distance of each other will depend on the specific arrangement of the points.

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It should be noted that 100,000 attendees for 10 bands a reasonable prediction as D. Yes, the pattern is linear. The average rate of change is about 10,000.

A straight-line link between two variables is referred to statistically as a linear relationship (or linear association).

A positive slope indicates a positive linear relationship, meaning that as one increases, the other also increases.

From the information, the year’s festival will feature 10 bands and the local TV station reports that the expected attendance is 100,000. The average rate will be:

= 100000 / 10

= 10000

The correct option is D.

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The derivative of an absolute value function is the slope of the tangent line to the curve at a specific point.

What is an absolute value function?

A function in algebra with the variable inside the absolute value bars is called an absolute value function. The most popular form of the absolute value function is f(x) = |x|, where x is a real integer, and this function is sometimes referred to as the modulus function.

The slope of the tangent line to the curve at a specific point is the derivative of an absolute value function, or for that matter, of any function. Such functions can be differentiated individually since they are piecewise functions.

Hence, the slope of the tangent line to the curve at a specific point is the derivative of an absolute value function.

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If we know the parameters a, b, and c in the first equation, we can find the equivalent parameters A, B, and C in the second equation by using the above relationships.

What is the logarithmic equation?

A logarithmic equation is an equation that involves the logarithm of an expression containing a variable. To solve exponential equations, first, see whether you can write both sides of the equation as powers of the same number.

The parameters in the two equations are related as follows:

A = log10(Psat/kPa) / ln(10)

B = b * ln(10)

C = c

T = t + 273.15 (to convert from Celsius to Kelvin)

Therefore, the relationship between the parameters in the two equations is:

A = log10(Psat/kPa) / ln(10)

B = b * ln(10)

C = c

T = t + 273.15

Hence, if we know the parameters a, b, and c in the first equation, we can find the equivalent parameters A, B, and C in the second equation by using the above relationships.

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Answer:

ar many types the total bar o

Answer:

I'm not sure if this is what you meant but if it's not, put a picture of the equation and I'll edit my answer

[tex]\frac{21}{670}[/tex]

Step-by-step explanation:

Assuming the equation is:

[tex]\frac{3}{5} * \frac{7}{2*67}[/tex]

The first thing you have to do is simplify:

2 * 67 = 134

[tex]\frac{3}{5} * \frac{7}{134}[/tex]

Multiply the numerators:

3 * 7 = 21

Then multiply the denominators:

5 * 134 = 670

Then simplify again:

[tex]\frac{21}{670}[/tex]

Step-by-step explanation:

For two triangles to be congruent, they either need to have three respectively equal sides (SSS), two respectively equal signs with the same angle between them (SAS), two same angles with a respectively equal side between them (ASA), or for a right-angled triangel, a respectively equal hypotenuse and side (RHS).

Here we have the triangles ΔABC and ΔCDE. We know DC is equal to BC because they are marked as such, same for AC and EC. We also know <ACB = <DCE because they are vertically opposite angles. Since we have two triangles with equal sides and the same angle between these sides (SAS), we can say these triangles are congruent.

The angle of separation between E and G is 12.0415945788 degrees.

Distance formula =[tex]$\sqrt{\left(\text { dif ferenceof } f^{\prime} x^{\prime} \text { points }\right)^2+\left(\text { Differenceof }{ }^{\prime} y^{\prime} \text { points }\right)^2} \\[/tex]

[tex]$& =\sqrt{(1-9)^2+(7-(-2))^2} \\[/tex]

[tex]& =\sqrt{(-8)^2+(9)^2} \\[/tex]

[tex]& =\sqrt{64+81} \\[/tex]

[tex]& =\sqrt{145} \\[/tex]

= 12.0415945788

As a result, the distance between E and G is 12.0415945788 miles.

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The composite functions f ∘ g and g ∘ f is defined as follows:

f ∘ g (x) = f(g(x))

g ∘ f (x) = g(f(x))

In other words, f ∘ g is the composition of two functions f and g, such that the output of g is used as the input to f. Similarly, g ∘ f is the composition of two functions g and f, such that the output of f is used as the input to g.

It's important to note that the order in which the functions are composed matters, as f ∘ g is not necessarily equal to g ∘ f. In general, the composite function of two functions is not commutative, meaning that the order in which the functions are composed affects the result.

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To develop a 95% confidence interval estimate of the population mean rating for Miami International Airport, the student would need to use the sample data obtained from the 50 business travelers. The steps to develop the confidence interval would include the following:

Calculate the sample mean: The mean of the sample data would be calculated by summing all of the ratings and dividing by the sample size (n = 50).

Calculate the standard deviation: The standard deviation of the sample data would be calculated to estimate the variability of the data.

Calculate the standard error: The standard error of the mean would be calculated by dividing the standard deviation by the square root of the sample size (n = 50).

Calculate the critical value: The critical value would be calculated based on the desired confidence level (95%) and the degrees of freedom (n - 1 = 49). The critical value would be used to determine the width of the confidence interval.

Develop the confidence interval: The lower and upper bounds of the confidence interval would be calculated by subtracting and adding the critical value to the sample mean, respectively.

The resulting confidence interval would give an estimate of the population mean rating for Miami International Airport with a 95% level of confidence. The interpretation of the confidence interval would be that if the sampling process were repeated many times, 95% of the resulting confidence intervals would contain the true population mean.

It is important to note that this process assumes that the sample was taken using simple random sampling techniques and that the data follows a normal distribution.

Therefore, The interpretation of the confidence interval would be that if the sampling process were repeated many times, 95% of the resulting confidence intervals would contain the true population mean.

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To develop a 95% confidence interval estimate of the population mean rating for Miami International Airport, the student would need to use the sample data obtained from the 50 business travelers. The steps to develop the confidence interval would include the following:

Calculate the sample mean: The mean of the sample data would be calculated by summing all of the ratings and dividing by the sample size (n = 50).

Calculate the standard deviation: The standard deviation of the sample data would be calculated to estimate the variability of the data.

Calculate the standard error: The standard error of the mean would be calculated by dividing the standard deviation by the square root of the sample size (n = 50).

Calculate the critical value: The critical value would be calculated based on the desired confidence level (95%) and the degrees of freedom (n - 1 = 49). The critical value would be used to determine the width of the confidence interval.

Develop the confidence interval: The lower and upper bounds of the confidence interval would be calculated by subtracting and adding the critical value to the sample mean, respectively.

The resulting confidence interval would give an estimate of the population mean rating for Miami International Airport with a 95% level of confidence. The interpretation of the confidence interval would be that if the sampling process were repeated many times, 95% of the resulting confidence intervals would contain the true population mean.

It is important to note that this process assumes that the sample was taken using simple random sampling techniques and that the data follows a normal distribution.

Therefore, The interpretation of the confidence interval would be that if the sampling process were repeated many times, 95% of the resulting confidence intervals would contain the true population mean.

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It took 245.33 watts of power to perform the work.

What is the work?

Work is defined as a force causing the movement or displacement of an object. In the case of a constant force, work is the scalar product of the force acting on an object and the displacement caused by that force.

The work done by Mr. Muscles can be calculated as follows:

Work = Force x Distance x Cos(θ)

where Force is 910 N, Distance is 4 m (since he lifts the weight 0.5 m 8 times), and θ is the angle between the force applied and the displacement of the weight, which is assumed to be 0° in this case since the weight is being lifted straight up.

Work = 910 N * 4 m * cos(0°) = 3640 N * m

The power required to perform this work in 15 seconds can be calculated as follows:

Power = Work / Time

Power = 3640 N * m / 15 s = 245.33 W

hence, it took 245.33 watts of power to perform the work.

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The line representing the equation part of the constraint [tex]$7 x_1+4 x_2 \leqslant 28$[/tex] goes through (4,0) & (0,7).

Hence, option (c) is the correct choice.

The constraint equation is g(x,y)=c, and we state that x and y are constrained by g(x,y)=c.

Constrained maximum and constrained minimum points are locations (x,y) that are maxima or minima of f(x,y) with the condition that they meet the constraint equation g(x,y)=c.

A constraint function can be translated into a new form that is equal to the original function; that is, the constraint boundary and feasible set for the problem remain unchanged, but the function's form does.

The given constraint is:

[tex]7 x_1+4 x_2 \leq 2[/tex]

If x_1=0

So, we will get:

4x_2=28

=>x=7

If x_2=0

7 x_1 =28

x_1 =4

So, the points are: (4,0)

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[tex]y = 4x^5 + Cx-6[/tex] the implicit solution to the initial value problem is y = [tex]4x^5 + Cx-6[/tex], where C is an arbitrary constant.

Start by isolating the y-derivative on the left side of the equation.

[tex]dy/dx = x^5 6 y^2[/tex]

Divide both sides by the coefficient of the y-derivative (6y2).

[tex]1/6(dy/dx) = x5/6 y2[/tex]

Integrate both sides with respect to y.

[tex]1/6∫(dy/dx)dy = ∫(x5/6 y2)dy 1/6y + C1 = (x5/6)y3/3 + C2[/tex]

Rearrange the equation to solve for y.

[tex]y = 4x5 + Cx-6[/tex]

Finally, the implicit solution to the initial value problem is [tex]y = 4x5 + Cx-6,[/tex]where C is an arbitrary constant.

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Answer:

$47

Step-by-step explanation:

First, we have to find 1% by taking

2.82 divided by 6 = $0.47

To find the original, we take

0.47 times 100 = $47

So, the original price of the sweatshirt is $47

The original price of the sweatshirt is, $47

A number or ratio that can be expressed as a fraction of 100 or a relative value indicating hundredth part of any quantity is called percentage.

To Calculate the percent of a number , divide the number by whole number and multiply by 100.

Given that;

Suppose you paid $2.82 sales tax for a sweatshirt in state where the sales tax is 6%.

Let the original price of the sweatshirt is, x

Hence, We can formulate;

6% of x = 2.82

6x / 100 = 2.82

6x = 2.82 x 100

6x = 282

x = 282 / 6

x = $47

Thus, The original price of the sweatshirt is, $47

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We can prove the equality using mathematical induction. We assume it holds for n and show it holds for n+1. Then, we add n+1 to both sides of the equation, proving the equality holds for n+1.

To prove the equality using mathematical induction, we must first assume it holds for n. Then, we can add n+1 to both sides of the equation:

∑ ― 1 = 1 ( 1 i(i 1)) + (n + 1) = 1 ― 1/ + (n + 1)

The right side of the equation is equal to 1 ― 1/ + n + 1, which simplifies to 1 ― 1/(n+1), proving the equality holds for n+1. Therefore, the equality holds for all n ≥ 2, and we have proven the equation using mathematical induction. Note: The expression in the equation should be n(n+1)/2, not 1/i(i+1).

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