Alice gardener wants to build a rectangular enclosure with a dividing fence in the middle of the rectangle. On one side, she plans to put some goats, on the other side she wants to raise some vegetables. The fence along the outside of the rectangle costs $3 per foot, but the dividing fence costs $12 per foot.



(a) Alice decides to spend $240 on the fencing, what is the maximum area she can enclose? Justify your answer.



(b) If Alice decided she wants to enclose 300 square feet, what is the minimum cost?

Answer:

The answer would be D.

(3,-5) is the original image.
To find your X, use the x from the first image and fill in the x which would be (3-1) which gives you (2,y)

to find Y, use the y from the first image and fill it it which is ( (-5) - 3 ) which gives you (x,-8)

therefore the full answer would be D. (2,-8)

Step-by-step explanation:

a. The elasticity function: E(x) = -8x²/(1536-2x²)

b. The elasticity at x = 20 is -2.78.

c. At x = 20, demand is elastic.

d. The value of x for which total revenue is a maximum is $12.

a. The elasticity function, E(x), can be calculated using the formula:

E(x) = (dQ/Q) / (dx/x)

where Q is the quantity demanded and x is the price. In this case, we have:

Q = D(x) = 1536 - 2x²

Taking the derivative with respect to x, we get:

dQ/dx = -4x

Using this, we can calculate the elasticity function:

E(x) = (dQ/Q) / (dx/x) = (-4x/(1536-2x²)) * (x/Q) = -8x²/(1536-2x²)

b. To find the elasticity at x = 20, we substitute x = 20 into the elasticity function:

E(20) = -8(20)²/(1536-2(20)²) = -3200/1152 = -2.78

So the elasticity at x = 20 is -2.78.

c. To determine whether demand is elastic, unit elastic, or inelastic at x = 20, we can use the following guidelines:

If E(x) > 1, demand is elastic.

If E(x) = 1, demand is unit elastic.

If E(x) < 1, demand is inelastic.

Since E(20) = -2.78, demand is elastic at x = 20.

d. To find the value(s) of x for which total revenue is a maximum, we use the formula for total revenue:

R(x) = xQ(x) = x(1536 - 2x²)

Taking the derivative of R(x) with respect to x, we get:

dR/dx = 1536 - 4x²

Setting this equal to zero to find the critical points, we get:

1536 - 4x² = 0

Solving for x, we get:

x = ±12

To determine whether these are maximum or minimum points, we take the second derivative of R(x):

d²R/dx² = -8x

At x = 12, we have d²R/dx² < 0, so R(x) is maximized at x = 12. Therefore, the value of x for which total revenue is a maximum is $12.

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The length of the missing side is equal to 18 inches.

In Mathematics and Geometry, the perimeter of a triangle can be calculated by using this mathematical equation:

P = a + b + c

Where:

By substituting the given parameters or dimensions into the formula for the perimeter of a triangle, we have the following;

51 = 18 + 15 + x

51 = 33 + x

x = 51 - 33

x = 18 inches.

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Answer:

-2

Step-by-step explanation:

7+9d=7d+3

7+2d=3

2d=-4

d=-2

The lateral surface area about x-axis is 114.1 square units.

To find the lateral surface area of the cone generated by revolving the line segment y=9/2x, 0≤x≤9 about the x-axis, we first need to find the length of the slant height of the cone.

We can think of the cone as being formed by rotating a right triangle about the x-axis.

The line segment y=9/2x intersects the x-axis at (0,0) and (9,81/2).

This forms a right triangle with base 9 and height √(81/2) = (9/2)√2.

The slant height of the cone is the hypotenuse of this right triangle, which can be found using the Pythagorean theorem:

l = √(9² + (9/2√2)²) = √(81 + 81/8) = (9/√2)√(9/8) = (9/2)√2

The lateral surface area of the cone can then be found using the formula:

L = πrl

where r is the radius of the base of the cone (which is equal to half the base of the right triangle, or 9/2) and

l is the slant height we just found.

Substituting in the values, we get:

L = π(9/2)(9/2)√2 = (81/4)π√2 ≈ 114.1

Therefore, the lateral surface area of the cone generated by revolving the line segment y=9/2x, 0≤x≤9 about the x-axis is approximately 114.1 square units.

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The cost of one blanket after calculations sums up as $32.

Let b be the cost of one blanket and s be the cost of one scarf in dollars. We can set up a system of equations based on the information given:

2b + 5s = 104

3b + 4s = 128

We want to solve for the cost of one blanket, so we'll solve for b in terms of s. We can start by multiplying the first equation by 3 and the second equation by 2 to create a system of equations where the coefficients of b will cancel each other out when we subtract the two equations:

6b + 15s = 312

6b + 8s = 256

Subtracting the second equation from the first, we get:

7s = 56

Dividing both sides by 7, we get:

s = 8

Now we can substitute s = 8 into either of the original equations to solve for b:

2b + 5(8) = 104

2b + 40 = 104

2b = 64

b = 32

Therefore, one blanket costs $32.

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The bases could be circled 40 times in 50 minutes at the same rate.

For this issue's solution, let's use unit rates.

In order to calculate the unit rate,

Considering that the player went 12 times around the bases in 15 minutes, the unit rate is 12/15, =  0.8 times per minute.

In a minute, the player would have circled the bases 0.8 times. By dividing the unit rate by the number of minutes, we can calculate how many times the bases could be circled in 50 minutes:

50 minutes x  0.8 times each minute = 40 times.

Therefore, the bases could be circled 40 times in 50 minutes at the same rate.

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The average amount of books read per student according to the histogram is given as follows:

1.27 books.

The mean of a data-set is given by the sum of all observations in the data-set divided by the number of observations, which is also called the cardinality of the data-set.

The histogram shows the number of times for each observation, hence:

Hence the mean is calculated as follows:

M = (7 x 0 + 9 x 1 + 6 x 2 + 4 x 3)/(7 + 9 + 6 + 4) = 1.27 books.

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We have evidence to suggest that the true percentage of college degrees in mathematics is different from 1%.

The null hypothesis is a type of hypothesis that explains the population parameter and is used to examine if the provided experimental data are reliable.

To test the claim that the percentage of college degrees in mathematics is not 1%, we can use a hypothesis test. Let's assume the null hypothesis is that the true percentage of college degrees in mathematics is 1%, and the alternative hypothesis is that it is different from 1%.

- Null hypothesis: The percentage of college degrees in mathematics is 1%.

- Alternative hypothesis: The percentage of college degrees in mathematics is different from 1%.

We can use a binomial distribution to model the number of graduates with math degrees in a sample of 12,317. Under the null hypothesis, the expected number of graduates with math degrees is:

Expected value = sample size * probability of math degrees = 12,317 * 0.01 = 123.17

Since we are testing at a 0.10 level of significance, the critical values for a two-tailed test are ±1.645 (using a standard normal distribution table).

The test statistic can be calculated as:

z = (observed value - expected value) / standard deviation

The standard deviation of the binomial distribution can be calculated as:

√(sample size * probability of success * (1 - probability of success))

So,

standard deviation = √(123.17 * 0.01 * 0.99) = 1.109

The observed value is 148.

The test statistic is:

z = (148 - 123.17) / 1.109 = 22.38

Since the absolute value of the test statistic is greater than 1.645, we can reject the null hypothesis at the 0.10 level of significance.

Therefore, we have evidence to suggest that the true percentage of college degrees in mathematics is different from 1%.

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The exact value of sin⁻¹(−1/2) is -π/6.

Given, sin⁻¹(-1/2)

The inverse sine function, sin⁻¹, or arcsin, returns the angle whose sine is equal to the given value. In this case, we are looking for the angle whose sine is -1/2.

Let y = sin⁻¹(-1/2)

sin (y) = -1/2

sin (y) = - sin (π/6)

sin (y) =  sin (- π/6)

y = - π/6

sin⁻¹(-1/2) = - π/6

To understand why the answer is -π/6, we can consider the unit circle. On the unit circle, the sine function represents the y-coordinate of a point corresponding to an angle. For -1/2, we need to find the angle where the y-coordinate is -1/2.

One such angle is -π/6, where the point on the unit circle is located in the fourth quadrant. At this angle, the y-coordinate is -1/2. Hence, sin⁻¹(−1/2) is -π/6.

Therefore, the exact value of sin⁻¹(−1/2) is -π/6.

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If 3 quarts is greater than 4prints, then the measure is not equivalent.

Equivalent units can be used to convert different units to the same unit for comparison. Equivalent means equal. For example , 1 kilogram is equal to 1,000 grams.

For example,

3 teaspoons = 1 tablespoon.

4 tablespoons = 1/4 cup.

5 tablespoons + 1 teaspoon = 1/3 cup.

8 tablespoons = 1/2 cup.

1 quart = 2pints

therefore 3 quarts = 2×3 = 6pints

therefore the statement that 3 quarter is greater than 4 prints is true and not an equivalent measure.

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Answer:

compounded continuously

Step-by-step explanation:

compounded continuously occurs more frequently than daily

After 6 hours, approximately 19.66 milligrams of the drug will remain in the patient's bloodstream.


To find the remaining amount of the drug in the patient's bloodstream after 6 hours, we'll use the decay function given: A(t) = P(1 - r)^t, where:

- A(t) is the remaining amount after t hours
- P is the initial amount (75 milligrams in this case)
- r is the decay rate per hour (20% or 0.20)
- t is the number of hours (6 hours)

Step 1: Plug in the given values into the formula.
A(t) = 75(1 - 0.20)^6

Step 2: Calculate the expression inside the parentheses.
1 - 0.20 = 0.80

Step 3: Replace the expression in the formula.
A(t) = 75(0.80)^6

Step 4: Raise 0.80 to the power of 6.
0.80^6 ≈ 0.2621

Step 5: Multiply the result by the initial amount.
A(t) = 75 × 0.2621 ≈ 19.66

So, approximately 19.66 milligrams of the drug will remain in the patient's bloodstream after 6 hours, rounded to the nearest hundredth.

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Answer: F, H, and J all have no real solution. The only equation that has a solution is

Step-by-step explanation: Use foil method.

The logistic function that represents the number of units occupied over time is given by:

[tex]N(t) = (K / (1 + A * e^(-r*t))),[/tex]

where N(t) is the number of units occupied at time t, K is the carrying capacity (maximum number of units that can be occupied),

A is the initial amount of units occupied, r is the growth rate, and e is the base of the natural logarithm.

In this case, the carrying capacity K is 1500 units, and the initial amount of occupied units A is 15 units. The growth rate r can be calculated as follows:

[tex]r = ln((10%)/(100% - 10%)) = ln(0.1/0.9) ≈ -0.101[/tex]

Substituting the given values into the logistic function, we get:

[tex]N(t) = (1500 / (1 + 15 * e^(-0.101*t)))[/tex]

Simplifying further, we get:

[tex]N(t) = (100 / (1 + e^(-0.101*t))) + 15[/tex]

Therefore, the logistic function that represents the number of units occupied over time is:

[tex]N(t) = (100 / (1 + e^(-0.101*t))) + 15[/tex], where t is measured in months.

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Answer:

The function g(x) = (3x)² can be simplified to g(x) = 9x², which is a vertical stretch of f(x) = x² by a factor of 9.

Therefore, the correct answer is B. The graph of g(x) is vertically stretched by a factor of 3 compared to the graph of f(x).

The original house has 70 square feet of flooring.

If the ratio of the miniature house to the original structure is 2:35, then we can say that the miniature house is 2/35th the size of the original house in terms of floor area. Let's assume that the original house has x square feet of flooring. Then, we can set up a proportion based on the ratios:

2/35 = 4/x

Solving for x, we get:

x = 70

Therefore if the ratio of ambers miniature house to the original structure is 2:35 and the miniature requires 4 square feet of the flooring then original house has 70 square feet of flooring.

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The probability that a randomly selected student prefers the arts or does not prefer literature is 8/11.

There are different ways to approach this problem, but one possible method is to use the concept of complement events.

First, we can calculate the probability of a randomly selected student preferring the arts. This is simply the proportion of students in the sample who prefer the arts, which is 9 out of 3+9+10 = 22. So, the probability is:

P(arts) = 9/22

Next, we can calculate the probability of a randomly selected student preferring literature. This is the proportion of students in the sample who prefer literature, which is 7+8 = 15 out of 22. So, the probability is:

P(literature) = 15/22

To find the probability of a student preferring the arts or not preferring literature, we can use the complement event that consists of students who do not prefer literature. This is the complement of the event "preferring literature", and its probability is:

P(not literature) = 1 - P(literature) = 1 - 15/22 = 7/22

Finally, we can use the addition rule for disjoint events (i.e., events that cannot occur at the same time) to calculate the probability of the event "preferring the arts or not preferring literature".

Since these events are not mutually exclusive (i.e., some students may prefer both the arts and literature), we need to subtract their intersection (i.e., students who prefer both) to avoid double-counting. Therefore, the probability is:

P(arts or not literature) = P(arts) + P(not literature) - P(arts and literature)

= 9/22 + 7/22 - 0

= 16/22

= 8/11

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equivalent integral with the order of integration reversed ST 2-3 F(x,y) dydc = o g(y) F(x,y) dedy+ So k(y) F(x,y) dardy Jh(v) a- he C- f(y) = g(y) = h(g) = k(y) = By reversing the order of integration, you've found an equivalent integral to the original one provided.

step-by-step explanation to achieve this, using the terms "integral," "reversed," and "equivalent" in the answer.

Step 1: Identify the original integral
The original integral is given as ∫∫ F(x, y) dy dx, where the integration limits are not explicitly provided. In this case, let's assume the limits of integration for y are from a(x) to b(x), and for x, they are from c to d.

Step 2: Sketch the region of integration
To reverse the order of integration, it's helpful to sketch the region of integration, which is the area in the xy-plane where the function F(x, y) is being integrated.

Step 3: Determine the new limits of integration
After sketching the region, determine the new limits of integration by considering the range of x for a given y value, and the range of y values. Let's assume the new limits for x are from g(y) to h(y), and for y, they are from e to f.

Step 4: Write the equivalent reversed integral
Now, you can write the equivalent integral with the order of integration reversed. In this case, it will be ∫∫ F(x, y) dx dy, with the new limits of integration. The complete reversed integral will look like:

∫(from e to f) [ ∫(from g(y) to h(y)) F(x, y) dx ] dy

By reversing the order of integration, you've found an equivalent integral to the original one provided.

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1) The maximum amount you should spend on rent is $840.

2) The maximum amount you should spend on rent is $570.

3) Your monthly income must be at least $2,733.33 to afford this apartment.

4) Your monthly income must be at least $3,000 to afford this apartment.

5) You need $1,980 to start renting the apartment.

1) With a monthly income of $2,800, the maximum amount you should spend on rent can be calculated using the 30% rule.
$2,800 x 0.30 = $840
So, the maximum amount you should spend on rent is $840.

2) With a monthly income of $1,900, the maximum amount you should spend on rent can be calculated using the 30% rule.
$1,900 x 0.30 = $570
So, the maximum amount you should spend on rent is $570.

3) To afford an apartment that rents for $820, your monthly income should be:
$820 ÷ 0.30 = $2,733.33
So, your monthly income must be at least $2,733.33 to afford this apartment.

4) To afford an apartment that rents for $900, your monthly income should be:
$900 ÷ 0.30 = $3,000
So, your monthly income must be at least $3,000 to afford this apartment.

5) To start renting an apartment that costs $665/month, you need the first and last month's rent, and a $650 security deposit.
First and last month's rent: $665 x 2 = $1,330
Total amount needed: $1,330 + $650 = $1,980
So, you need $1,980 to start renting the apartment.

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The probability that the outcome is a divisor of 4 given that it is prime is 0.125, or 12.5%.

Since we are given the probabilities of different outcomes, we can use the definition of conditional probability to find p(a|b), which represents the probability that the outcome is a divisor of 4 given that it is prime.

The formula for conditional probability is:

p(a|b) = p(a ∩ b) / p(b)

where p(a ∩ b) represents the probability of both events happening simultaneously.

Looking at the table of outcomes and their probabilities, we can see that there are four prime numbers: 2, 3, 5, and 7. Of these, only 2 is a divisor of 4.

Therefore, p(a ∩ b) is the probability that the outcome is 2, which is 0.1.

The probability of the outcome being prime is the sum of the probabilities of the four prime outcomes, which is:

p(b) = 0.1 + 0.2 + 0.3 + 0.2 = 0.8

Substituting these values into the formula for conditional probability, we get:

p(a|b) = p(a ∩ b) / p(b) = 0.1 / 0.8 = 0.125

Therefore, the probability that the outcome is a divisor of 4 given that it is prime is 0.125, or 12.5%.

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The function has a local maximum at x = -6.

To determine the intervals on which the function f(x) = 4x³ - 12x² - 576x + 6 is increasing or decreasing, we first find its derivative, f'(x), and then analyze its critical points.

f'(x) = 12x² - 24x - 576

Now, set f'(x) = 0 and solve for x:

12x² - 24x - 576 = 0

Divide by 12:
x² - 2x - 48 = 0

Factor:
(x - 8)(x + 6) = 0

So, the critical points are x = 8 and x = -6.

Analyze the intervals:
f'(-7) > 0, so increasing on (-∞, -6)
f'(0) < 0, so decreasing on (-6, 8)
f'(9) > 0, so increasing on (8, ∞)

The function f(x) is decreasing on the interval (-6, 8). It is increasing on the interval (-∞, -6) and the interval (8, ∞). The function has a local maximum at x = -6.

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Joe would need to successfully kick 11 consecutive field goals to raise his success rate to 49%.

Let's use the given terms and solve the problem step by step.

1. Joe's current success rate: He made 7 out of 26 field goal kicks.
2. Desired success rate: 49%

Let's use 'x' as the number of consecutive field goals Joe needs to make to reach a 49% success rate.

Step 1: Calculate the total number of kicks after making 'x' consecutive goals.
Total kicks = 26 (previous kicks) + x (consecutive goals)

Step 2: Calculate the total number of successful kicks after making 'x' consecutive goals.
Successful kicks = 7 (previous successful kicks) + x (consecutive successful goals)

Step 3: Calculate the success rate (total successful kicks / total kicks) and set it equal to 49%.
(Successful kicks / Total kicks) = 49/100

Step 4: Substitute the expressions from Steps 1 and 2 into the equation from Step 3.
(7 + x) / (26 + x) = 49/100

Step 5: Solve for 'x'.
49 * (26 + x) = 100 * (7 + x)

1274 + 49x = 700 + 100x
49x - 100x = 700 - 1274
-51x = -574

x = 574 / 51
x ≈ 11.25

Since Joe cannot make a fraction of a goal, he needs to make 12 consecutive field goals to reach a success rate of at least 49%.

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We can use the Pythagorean theorem to solve this problem, which states that in a right triangle, the square of the hypotenuse (the longest side) is equal to the sum of the squares of the other two sides.

Let x be the length of the slide. Then we have a right triangle with legs of length 5 (the height of the ladder) and x, and hypotenuse of length 12 (the distance from the ladder to the bottom of the slide).

Using the Pythagorean theorem:

12^2 = 5^2 + x^2

144 = 25 + x^2

Subtracting 25 from both sides:

119 = x^2

Taking the square root of both sides:

x ≈ 10.91

Therefore, the slide needs to be about 10.91 feet long.

The value of S2 is 50, under the condition that {sn} is  a geometric sequence that starts with an initial index of 0.

Here we have to apply the principles of geometric progression.
The derived formula for regarding the nth term concerning the geometric sequence is
[tex]= ar^{n-1 }[/tex]
Here
a = first term and r is the common ratio.
For the given case from the question
a = 2
r = 5.
Then,
s2 = a× r²
= 2×5²
= 50.
A geometric sequence refers to a particular sequence of numbers that compromises each term after the first is evaluated by multiplying the previous one by a fixed one , non-zero number known as  the common ratio.

For instance, if the first term of a geometric sequence is 2 and the common ratio is 5, then the sequence would be 2, 10, 50, 250.
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20 pieces of 10 5/6 inch bar can be cut from a stock 29 foot bar.

To calculate the number of pieces of 10 5/6 inch bar that can be cut from a 29 foot bar, we need to first convert the measurements to a common unit. One foot is equal to 12 inches, so 29 feet equals 348 inches.

Next, we need to determine how many 10 5/6 inch bars can be cut from the 348-inch stock bar. To do this, we can use division. First, we need to convert the mixed number 10 5/6 to an improper fraction by multiplying the whole number by the denominator and adding the numerator. This gives us 125/6 inches.

Now, we can divide the length of the stock bar (348 inches) by the length of one 10 5/6 inch bar (125/6 inches). This gives us:

348 / (125/6) = 20.736

Since we cannot cut a partial bar, we need to round down to the nearest whole number. Therefore, we can cut 20 pieces of 10 5/6 inch bar from a 29 foot stock bar.

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