Which expressions represent the length of side MN?



Choose 2 answers:



A. 4 • sin (65)



B. 1. 9 • cos (65)



C. 4/sin (65)



D. 1. 9/cos (65)



PLEASE HELP!!

The critical point is c = 2, the function is decreasing on the interval 0 < x < 2, and increasing on the interval x > 2.

To find the critical point of the function y = x^2 - 4x, we first need to find its derivative, which represents the slope of the tangent line at any point on the curve.

The derivative of y with respect to x is:

y' = 2x - 4

Now, we need to find the critical points, which occur where the derivative is zero or undefined. In this case, the derivative is a polynomial, so it is never undefined. To find where it equals zero, we set y' equal to zero:

0 = 2x - 4

Solving for x, we get:

x = 4/2 = 2

So, the critical point is c = 2.

Now, we need to determine if the function is increasing or decreasing on the interval x > 0. To do this, we can analyze the sign of the derivative. If y' > 0, the function is increasing; if y' < 0, the function is decreasing.

For x > 2 (to the right of the critical point), the derivative y' = 2x - 4 is positive (since 2x > 4 when x > 2). Therefore, the function is increasing on the interval x > 2.

For x < 2 (to the left of the critical point), the derivative y' = 2x - 4 is negative (since 2x < 4 when x < 2). Therefore, the function is decreasing on the interval 0 < x < 2.

In summary, the critical point is c = 2, the function is decreasing on the interval 0 < x < 2, and increasing on the interval x > 2.

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If the mean time for problem 1 is 10 minutes, and the mean time for problem 2 is 15 minutes, The dot plots show the amount of time it takes each person in a random sample to complete two similar problems.

To find the mean time for each problem, we need to add up all the times and divide by the total number of people in the sample. Let's assume that the first dot plot represents problem 1 and the second dot plot represents problem 2.

After calculating the mean times for each problem, we can make a comparative inference based on the mean values. For instance, if the mean time for problem 1 is 10 minutes, and the mean time for problem 2 is 15 minutes, we can infer that problem 2 takes longer to complete on average than problem 1.

Comparative inference refers to the process of comparing two or more sets of data to draw conclusions about their similarities or differences. In this case, we are comparing the mean times for two similar problems to see which one takes longer on average.

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The probability of the next coat being sold as a medium navy coat is 11 / 95 when a store sells a coat in three sizes: small, medium, and large. The coat comes in red, navy, and tan.

We need to find the probability that the next coat sold as a medium navy coat. To find the probability we need to find the total number of coats and the number of medium navy coats,

Given data:

Medium navy coat = 22

Total Number of small coats = 18 + 24 +19 = 61

Total Number of medium coats = 21 + 22 + 25 = 68

Total Number of large coats = 19 + 20 + 22 = 61

From the given data the total number of coats is = 61 + 68 + 61 = 190

The probability that the next coat sold as a medium navy coat = a number of medium navy coats / total number of coats.

= 22 / 190

= 11 / 95

Therefore, the probability of the next coat being sold as a medium navy coat is  11 / 95

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Ms. Applebee's apple is because they lacked the necessary human capital to understand the play on words.

To begin with, let's define what we mean by capital. In economics, capital refers to any resource that can be used to produce goods or services. This can include physical capital, such as machinery and equipment, as well as financial capital, such as money and investments.

However, there is another type of capital that is just as important: human capital. Human capital refers to the skills, knowledge, and abilities that individuals possess, which can be used to create economic value. This can include things like education, training, and experience.

This is where the concept of human capital comes in. In order to "see" Ms. Applebee's apple (or capital), you need to have the skills and knowledge to understand the play on words. This requires a certain level of human capital - specifically, linguistic and cultural capital.

Linguistic capital refers to the ability to use language effectively, including understanding and interpreting language in different contexts. Cultural capital, on the other hand, refers to the knowledge and skills needed to navigate different cultural contexts and understand cultural references.

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The expected value of x is given as follows:

E(X) = 1.6.

The expected value of a discrete distribution is given by the sum of each outcome multiplied by it's respective probability.

The distribution for this problem is given as follows:

Hence the expected value is given as follows:

E(X) = -7 x 0.2 - 3 x 0.1 + 3 x 0.4 + 7 x 0.3

E(X) = 1.6.

The table is given by the image presented at the end of the answer.

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The value of x as an improper fraction in its simplest form is 14/5.

To solve the inequality 5x - 4 < 10, we need to isolate x on one side of the inequality. First, we add 4 to both sides:

5x - 4 + 4 < 10 + 4

5x < 14

Then, we divide both sides by 5:

5x/5 < 14/5

x < 2.8

Therefore, the solution to the inequality is x < 2.8. However, the question asks for the answer as an improper fraction in its simplest form. To convert 2.8 to an improper fraction, we multiply both the numerator and denominator by 10 to get rid of the decimal:

2.8 * 10 / 1 * 10 = 28 / 10

To simplify the fraction, we divide both the numerator and denominator by their greatest common factor, which is 2:

28 / 10 = 14 / 5

Therefore, the answer is 14/5.

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The surface area of octagonal pyramid with height of 12 and a side length of 4.24 is 281.477 unit²

Height of octagonal pyramid = 12

Side length of octagonal pyramid = 4.14

The surface area of octagonal pyramid is  

SA = 2s²( 1 + √2) + 4sh

Here, s is side length of the octagonal pyramid = 4.14

h is height of the octagonal pyramid = 12

putting the values in the equation we get

SA = 2 × 4.14 ( 1 + √2 ) + 4 × 4.14 × 12

SA = 82.757 + 198.72

SA = 281.477

The surface area of octagonal pyramid is 281.477

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We know that the total mass of 5 polished stones would be 7 grams according to this equation

Sure! The equation m = 1.4p represents the mass, m, in grams, of p polished stones.

This means that if you know the number of polished stones, p, you can use this equation to calculate their total mass, m, in grams. For example, if you have 5 polished stones, you can plug in p = 5 and solve for m: m = 1.4 x 5 = 7 grams.

So the total mass of 5 polished stones would be 7 grams according to this equation.

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Based on the equation, the number of grams which the mass increase for every 7 polished stones is: D. 9.8 g.

In Mathematics and Geometry, the slope-intercept form of the equation of a straight line is given by this mathematical equation;

y = mx + c

Where:

Based on the information provided above, a linear equation that models the mass in grams is given by;

y = mx + c

m = 1.4p

By substituting the given parameter (x = 7 polished stones), we have the following:

m = 1.4(7)

m = 9.8 g

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Missing information:

The question is incomplete and the complete question is shown in the attached picture.

The distance between the tip of the ladder to the ground or the value of 'x' is 13.27 ft.

We know that the ladder is leaning on the wall and thus it makes a right-angle triangle, where:

the hypotenuse(h) is the length of the ladder,

the base(b) is the distance between the foot of the ladder and the bottom of the wall,

and the height(x) is the distance between the tip of the ladder to the bottom of the wall which we need to find.

As the question is on right angled triangle we can use the Pythagoras theorem to find the value of 'x':

[tex]Height^2 + Base^2 = Hypotenuse^2\\x^2 + b^2 = h^2[/tex]

Now we know that h= 15ft, and b=7ft.

Substituting the values in the above equation we get :

[tex]x^2 + 7^2 = 15^2\\x^2 = 225 - 49\\x = \sqrt{176}\\x= 13.27 ft.[/tex]

Therefore the distance between the tip of the ladder to the ground or the value of 'x' is 13.27 ft.

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Using the coupon that offers a $0.50 discount on the larger package would yield the lowest price per ounce at $0.1875. Kelsey should use this coupon to get the best value for her favorite crackers.

Kelsey has two options when it comes to purchasing her favorite crackers, and she needs to determine which coupon will result in the lowest price per ounce. To make an informed decision, Kelsey should compare the price per ounce of both cracker sizes and apply the appropriate coupon accordingly.

First, Kelsey should find the price per ounce for each size by dividing the total price of the package by the total number of ounces in the package. For example, if the smaller package costs $2.00 and contains 8 ounces of crackers, the price per ounce would be $2.00 / 8 = $0.25 per ounce. Similarly, if the larger package costs $3.50 and contains 16 ounces, the price per ounce would be $3.50 / 16 = $0.21875 per ounce.

Next, Kelsey should determine the discount offered by each coupon and calculate the new price per ounce after applying the respective coupon. For instance, if one coupon provides a 10% discount on the smaller package, the new price per ounce would be $0.25 * (1 - 0.1) = $0.225 per ounce. If another coupon offers a $0.50 discount on the larger package, the new price per ounce would be ($3.50 - $0.50) / 16 = $0.1875 per ounce.

Finally, Kelsey should compare the adjusted price per ounce for both packages and select the coupon that results in the lowest price per ounce. In this example, using the coupon that offers a $0.50 discount on the larger package would yield the lowest price per ounce at $0.1875. Therefore, Kelsey should use this coupon to get the best value for her favorite crackers.

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If the height of a model of the Cape Hatteras Lighthouse given that the scale used is 8 inches to 30 feet. This means that for every 8 inches on the model, the actual lighthouse is 30 feet tall. Therefore, the height of the model is 4.67 feet.

The actual lighthouse is 210 feet tall. To find the height of the model, we need to use a proportion. We can set up the proportion as:

8 inches / 30 feet = x inches / 210 feet

We can cross-multiply to get:

8 inches * 210 feet = 30 feet * x inches

Simplifying, we get:

x inches = (8 inches * 210 feet) / 30 feet = 56 inches

However, the question asks for the height of the model in feet, so we need to convert 56 inches to feet by dividing by 12:

56 inches / 12 inches per foot = 4.67 feet

Therefore, the height of the model is 4.67 feet.

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The equation of the line passing through point P and parallel to line A is

y = (5/7)x + 2

Line A passed through points (0, -1) and (7, 4), hence equation of line passing through point (0, -1) and (7, 4).

find the slope of the line:

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

m = (4 - (-1)) / (7 - 0)

m = 5/7

use the point-slope form of the equation of a line with the point (0, -1):

y - (-1) = (5/7)(x - 0)

y + 1 = (5/7)x

y = (5/7)x - 1

Therefore, the equation of the line passing through the points (0, -1) and (7, 4) is y = (5/7)x - 1.

for a line parallel to line A passing through point P we change the intercept to give

y = (5/7)x + 2

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The annual interest rate the third party will receive for the investment is approximately 7.7%.

After buying a new car, you sold your old car and took a 180-day note for $4,500 at 7.5% simple interest as payment. The principal plus interest will be due at the end of 180 days.

First, let's calculate the total amount due at the end of the 180 days. To do this, we'll use the formula for simple interest:

Interest = Principal x Rate x Time
Interest = $4,500 x 7.5% x (180/360) [As it's for half a year]
Interest = $4,500 x 0.075 x 0.5
Interest = $168.75

Now, let's add the interest to the principal to find the total amount due:
Total Amount = Principal + Interest
Total Amount = $4,500 + $168.75
Total Amount = $4,668.75

Sixty days later, you sell the note to a third party for $4,550. The third party will receive the remaining interest for 120 days. We need to find the annual interest rate for the third party's investment.

First, let's find the remaining interest:
Remaining Interest = Total Amount - $4,550
Remaining Interest = $4,668.75 - $4,550
Remaining Interest = $118.75

Now, let's find the annual interest rate for the third party's investment using the simple interest formula, but solving for the rate:

Rate = (Remaining Interest) / (Principal x Time)
Rate = $118.75 / ($4,550 x (120/360))
Rate = $118.75 / ($4,550 x 0.3333)

Rate ≈ 0.077 (approximated to 3 decimal places)

So, the annual interest rate the third party will receive for the investment is approximately 7.7%.

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Check the picture below.

Answer:

The length for which 98% of the nails will be longer is approximately 6.466 cm.

Step-by-step explanation:

First, we need to convert the units of the standard deviation to centimeters, since the mean is also given in centimeters. 2 mm is equal to 0.2 cm.

Next, we need to find the z-score that corresponds to the 98th percentile. We can use a standard normal distribution table or calculator to find this value. The z-score corresponding to the 98th percentile is approximately 2.33.

Finally, we can use the formula for a z-score to find the length of nail corresponding to this z-score:

where:

Solving for x, we get:

Therefore, the length for which 98% of the nails will be longer is approximately 6.466 cm.

The number of different sandwiches that can be created with two different meats is D. 6.

The number of different sandwiches that can be created with two different meats can be found by using the combination formula: nCr = n! / r!(n-r)!

In this case, we have 4 options for the first meat and 3 options for the second meat (since we cannot repeat the first meat). Therefore, the number of different sandwiches is:

4C2 = 4! / 2!(4-2)! = 6

So there are 6 different sandwiches that can be created with two different meats.

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To integrate the function ∫(x² - 36) dx, we first need to factor out the expression inside the parentheses:
∫(x² - 36) dx = ∫(x - 6)(x + 6) dx

We can then use the power rule of integration to find the antiderivative:
∫(x - 6)(x + 6) dx = (1/3)x³ - 6x + C, where C is the constant of integration.

Since the original problem states X > 6, we can evaluate the definite integral using these limits:
∫(x² - 36) dx from 6 to X = [(1/3)X³ - 6X] - [(1/3)(6)³ - 6(6)]
= (1/3)X³ - 6X - 68

Therefore, the answer in exact form is (1/3)X³ - 6X - 68.

To integrate the given function, first note the correct notation for the function: ∫(x^2)/(x^2 - 36) dx for x > 6.

To solve this, we can use partial fraction decomposition. The given function can be rewritten as:

∫(A(x - 6) + B(x + 6))/(x^2 - 36) dx

Solving for A and B, we find that A = 1/12 and B = -1/12. Now we rewrite the integral as:

∫[(1/12)(x - 6) - (1/12)(x + 6)]/(x^2 - 36) dx

Next, separate the two terms and integrate them individually:

(1/12)∫[(x - 6)]/(x^2 - 36) dx - (1/12)∫[(x + 6)]/(x^2 - 36) dx

Now, notice that the integrals are of the form ∫u'/u dx. The integral of this form is ln|u|. So we have:

(1/12)[ln|(x - 6)| - ln|(x + 6)|] + C

Using the logarithm property, we can rewrite the answer as:

(1/12)ln|((x - 6)/(x + 6))| + C

That is the exact form of the antiderivative for the given function.

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If the area of a triangle is half of its base times height then the reasonable value of x is = 4.

A triangle-shaped table top with an area of 324 square inches has a base of 8x+4 inches and a height of 4x+2 inches. Given that the area of a triangle is half of its base times height, we can use the formula:

Area = (1/2) * base * height

Plug in the given values:

324 = (1/2) * (8x + 4) * (4x + 2)

Now, we need to solve for x. Follow these steps:

1. Multiply both sides of the equation by 2 to get rid of the fraction:

2 * 324 = (8x + 4) * (4x + 2)

2. Simplify the equation:

648 = (8x + 4) * (4x + 2)

3. Expand the equation by multiplying the terms in the parentheses:

648 = 32x^2 + 16x + 16x + 8

4. Combine like terms:

648 = 32x^2 + 32x + 8

5. Move all terms to one side of the equation to set it equal to zero:

32x^2 + 32x - 640 = 0

Now, we need to find a reasonable value of x. You can solve this quadratic equation using factoring, the quadratic formula, or a graphing calculator. Using a graphing calculator, we find that x is approximately 4.

Therefore, a reasonable value of x in this situation is 4.

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The given statement "A function f(x) = 3x^4 dominates g(x) = x^4" is True, which means that as x gets larger, the value of f(x) will increase much more rapidly than the value of g(x).

As x increases or decreases, the 3x^4 term in f(x) will grow faster or be larger in magnitude than the x^4 term in g(x). Since f(x) grows faster or has a larger magnitude than g(x), we can conclude that f(x) dominates g(x).

Therefore, the function f(x) = 3x^4 has a higher degree than g(x) = x^4

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The height of a cone with a diameter of 12m whose volume is 226m³ is 6 meters.

The formula for the volume of a cone is

V = (1/3) * π * r^2 * h

where V is the volume, r is the radius, h is the height, and π is approximately equal to 3.14.

We know the diameter of the cone is 12m, which means the radius is 6m.

We also know that the volume of the cone is 226m^3.

Substituting these values into the formula, we get:

226 = (1/3) * π * 6^2 * h

Simplifying:

226 = (1/3) * 3.14 * 36 * h

226 = 37.68h

h = 226/37.68

h ≈ 6

Therefore, the height of the cone is approximately 6 meters.

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The number of minutes you can use the card is 9 minutes

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

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

f(x) = 0.25x + 0.75

If the total cost of the card is $5.00, the number of minutes you is

0.25x + 0.75 = 5

So, we have

0.25x = 4.25

Divide by 0.25

x = 9

Hence, the number of minutes is 9

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Answer:

2 inches

Step-by-step explanation:

The area of the metal plate can be calculated by subtracting the area of the hole from the area of the original plate. The area of the original plate is:

8 inches x 4 inches = 32 square inches

The area of the hole is:

8 inches x 4 inches = 32 square inches

So the area of the metal that remains is:

32 square inches - 32 square inches = 0 square inches

According to the equation given, we know that:

(8 + 2x)(4 + 2x) - 32 = 32

Expanding this equation we get:

32 + 16x + 8x + 4x^2 - 32 = 32

Simplifying and rearranging we get:

4x^2 + 24x - 32 = 0

Dividing both sides by 4 we get:

x^2 + 6x - 8 = 0

We can solve this quadratic equation by factoring:

(x + 4)(x - 2) = 0

So x = -4 or x = 2. Since the width of the frame cannot be negative, the only valid solution is x = 2.

Therefore, the frame width is 2 inches.

Now we need to apply the given boundary conditions to obtain the specific solution for u(x, y):

Boundary conditions in x-direction:
X(2) = X(27) = 0

Boundary conditions in y-direction:
Y(0) = Y(2m) = 0

1. Identify the Poisson equation and boundary conditions.
2. Use the method of separation of variables to solve the equation.
3. Apply the boundary conditions to obtain the specific solution.

Step 1: Identify the Poisson equation and boundary conditions
The given Poisson equation is:
Δu + 2g = 27 - 1,
where Δu is the Laplacian of the potential function u(x, y).

The provided boundary conditions are:
u(2, 0) = u(2, 2m) = 0,
u(0, y) = u(27, y) = 0.

Step 2: Use the method of separation of variables
We assume that the solution u(x, y) can be written as a product of two functions, one depending on x and the other depending on y, i.e., u(x, y) = X(x)Y(y).

Now, let's substitute this into the Poisson equation:
Δu + 2g = 27 - 1,
which becomes
(X''(x)/X(x) + Y''(y)/Y(y)) + 2g = 26.

Separate the variables:
X''(x)/X(x) = -Y''(y)/Y(y) - 2g = λ,
where λ is the separation constant.

This gives us two ordinary differential equations:
X''(x) = λX(x),
Y''(y) = -(λ + 2g)Y(y).

Step 3: Apply the boundary conditions
Now we need to apply the given boundary conditions to obtain the specific solution for u(x, y):

Boundary conditions in x-direction:
X(2) = X(27) = 0

Boundary conditions in y-direction:
Y(0) = Y(2m) = 0

Solving these equations with their respective boundary conditions will give us a specific solution for the potential function u(x, y). However, it is important to note that solving these equations involves solving eigenvalue problems and possibly infinite series expansions. The full solution process is quite involved and goes beyond the scope of this answer.

Nevertheless, I hope this outline of the solution method helps you understand the process of solving the Poisson equation with given boundary conditions.

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The derivative of y=f(x) = x⁴ - 7x + 5 is dy/dx = 4x³ - 7. For x = 5 and Δx=0.2, dy = 1.986 and Δy = -54.5504.

Given the function y = f(x) = x⁴ - 7x + 5, we can find its derivative with respect to x using the power rule of differentiation:

dy/dx = d/dx(x⁴) - d/dx(7x) + d/dx(5) = 4x³ - 7

Now, we can use the given value of x = 5 and Δx = 0.2 to find the values of dy and Δy:

dy = (4x³ - 7) dx, evaluated at x = 5 and Δx = 0.2

dy = (4(5)³ - 7) (0.2) = 198.6 × 10^(-2)

This means that a small change of 0.2 in x results in a change of about 1.986 in y.

To find Δy, we use the formula:

Δy = f(x + Δx) - f(x)

Substituting x = 5 and Δx = 0.2, we get:

Δy = ((5 + 0.2)⁴ - 7(5 + 0.2) + 5) - (5⁴ - 7(5) + 5)

Simplifying this expression gives:

Δy = (122.4496 - 177) = -54.5504

This means that a small change of 0.2 in x results in a change of about -54.5504 in y.

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Answer: 36

Step-by-step explanation:

[tex]\frac{1}{3} \pi 15^{2} h=2700\pi \\75h=2700\\h=36[/tex]

The training program with the least rate of change is Rachel's, with an increase of 3 minutes per day.

Rachel:
Starting minutes: 30
Increase in rate: 3 minutes per day
Equation: f(x) = 3x + 30

Nicole:
Starting minutes: 30 (since f(0) = 5(0) + 30 = 30)
Increase in rate: 5 minutes per day
Equation: f(x) = 5x + 30

To find the training program with the least rate of change, we need to find the derivative of each equation and set it equal to zero:

f'(x) = 3 for Rachel's equation
f'(x) = 5 for Nicole's equation

Since 3 is less than 5, Rachel's training program has the least rate of change. Therefore, the rate of change in minutes per day for Rachel's training program that has the least rate of change is 3 minutes per day.

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B. Whole number enlarges, fraction reduces.

When a figure is dilated by a whole number, the image is enlarged by a factor of that whole number. For example, if a figure is dilated by a scale factor of 2, the image will be twice as large as the original.

On the other hand, when a figure is dilated by a fraction, the image is reduced. For example, if a figure is dilated by a scale factor of 1/2, the image will be half as large as the original.

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From the margin gain formula, the number of hours he should have spent working on problems are equal to 2 hours and number of hours he should have spent reading are equal to other 2 hours. So, option (c) is correct one.

The meaning of ''margin'' is either the ''edge'' or the last unit and it is calculated by the incremental adjustment to the outcome, due to a unit change in the control variable. Marginal gain = Ratio of change in the outcome variable to the change in the control variable.

We have, total number of practice questions raises a students exam score

= 25

Along with practice questions the same amount as reading the textbook for 1 hour. In this problem, the outcome variable is the number of practice problems solved and the control variable is the number of hours Eric spent working on the practice problems. The above figure 1 table shows the total number of problems solved and time. First we determine the marginal gain of each hour Eric spent working on the practice questions. See the table present in above figures 2. Now, he has only 4 hours of study time for the best exam score as possible. We have to determine number of hours he should have spent reading and working on problems. It is assumed that students always cover the same number of pages during each hour they read the textbook so the advice provide by their teaching assistant that they can establish the relationship between time spent on reading the textbook and doing practice problems. The relation is below, 1 hour of reading the textbook = 15 practice problems solved. So, we can compare the the effectiveness of Eric's time spent on either working on practice problems or reading the textbook by using the table above figure 3. The decision rule for the optimal allocation of Eric's 4 hours of work is, If he can solve more than 15 practice problems in any of the 4 hours, then he should spend that particular hour working on the practice problems instead of reading the textbook. If not, then Eric should spend that hour to read the textbook instead.

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Complete question:

The table present in above figure 1 complete the question.

Eric is a hard-working college freshman. One Sunday, he decides to work nonstop until he has answered 100 practice problems for his math course. He starts work at 8:00 AM and uses a table to keep track of his progress throughout the day. He notices that as he gets tired, it takes him longer to solve each problem. later, the teaching assistant in charless economics course gives him some advice. the teaching assistant says, based on past experience, working on 25 problems raises a students exam score by about the same amount as reading the textbook for 1 hour. for simplicity, assume students always cover the same number of pages during each hour they spend reading. given this information, in order to use his 4 hours of study time to get the best exam score possible, how many hours should he have spent working on problems and how many should he have spent reading?

a) 1 hour working on problems, 3 hours reading

b) 2 hours working on problems, 2 hours reading

c) 3 hours working on problems, 1 hour reading

d) 4 hours working on problems, 0 hours reading

The point(s) on the curve y = x²/6 closest to the point (0,0) are (0,0) and (±√2, 2/3).

To find the point(s) on the curve y = x²/6 closest to the point (0,0), we can use the distance formula between two points:

d = √((x₁ - x₂)² + (y₁ - y₂)²)

where (x₁, y₁) is a point on the curve and (x₂, y₂) is the point (0,0).

We want to minimize the distance d, which is equivalent to minimizing d². Therefore, we can minimize:

d² = (x₁ - 0)² + (y₁ - 0)²

= x₁² + y₁²

subject to the constraint that the point (x₁, y₁) is on the curve y = x²/6.

Substituting y = x²/6 into the expression for d², we get:

d² = x₁² + (x₁²/6)

= (7/6)x₁²

To minimize d², we minimize x₁². Since x₁² is always non-negative, the minimum occurs when x₁² = 0 or when the derivative of d² with respect to x₁ is zero.

Taking the derivative of d² with respect to x₁, we get:

d²/dx₁ = (7/3)x₁

Setting this equal to zero, we get x₁ = 0.

Therefore, the point (0,0) is one of the closest points on the curve to the point (0,0).

To find the other closest point(s), we can solve y = x²/6 for x² and substitute it into the expression for d²:

x² = 6y

d² = 7x²/6 = 7y

Therefore, to minimize d², we need to minimize y. Since y is always non-negative, the minimum occurs when y = 0 or when the derivative of d² with respect to y is zero.

Taking the derivative of d² with respect to y, we get:

d²/dy = 7

Setting this equal to zero, we get y = 0.

Substituting y = 0 into y = x²/6, we get x = 0. Therefore, the point (0,0) is one of the closest points on the curve to the point (0,0).

To find the other closest point, we can solve y = x²/6 for x:

x² = 6y

x = ±√(6y)

Substituting this into the equation for y, we get:

y = (√(6y))²/6 = 2/3

Therefore, the other closest points are (±√2, 2/3).

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