Find a Cartesian equation for the curve and identify it. r2cos(2θ)=1 a. ellipse b. parabola c. circle d. hyperbola e. limaçon

Based on the survey results, a reasonable estimate of the number of computers owned by residents in the town is between 2.02 (2.42 - 0.4) and 2.82 (2.42 + 0.4). The margin of error of ±0.4 indicates that the estimate is likely to be within this range.

Sure! So, the survey of 120 households in the town found that the average number of computers in a household was 2.42. However, because this was a sample survey and not a complete census of all households in the town, there is some level of uncertainty in the estimate.

The margin of error of +-0.4 means that we can be 95% confident interval that the true average number of computers in households in the town falls within the range of 2.02 (2.42 - 0.4) and 2.82 (2.42 + 0.4).

So, a reasonable estimate of the number of computers owned by residents in the town would be around 2.42, but with a range of 2.02 to 2.82.

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Answer:

AB ≈ 12.6

Step-by-step explanation:

the length of arc AB is calculated as

AB = circumference of circle × fraction of circle

     = 2πr × [tex]\frac{45}{360}[/tex] ( r is the radius )

    = 2π × 16 × [tex]\frac{1}{8}[/tex]

    = 32π × [tex]\frac{1}{8}[/tex] ( cancel 8 and 32 by 8 )

   = 4π

  ≈ 12.6 ( to the nearest tenth )

Answer: [tex]F= \frac{G-H}{n}[/tex]

Step-by-step explanation:

I just isolated F by dividing both sides by n.

Mr. Stevenson will need to cover approximately 314 square feet of the patio with concrete sealer.

To calculate the area of a circle, we can use the formula:

Area = π * r^2

where π (pi) is around 3.14, and r is the radius of the circle. In this example, the radius is 10 feet.

Area = 3.14 * (10 ft)^2

Area = 3.14 * 100 sq ft

Area ≈ 314 sq ft

So, Mr. Stevenson will need to cover approximately 314 square feet of the patio with concrete sealer.

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What is the area of a circular patio with a radius of 10 feet, using the approximation of pi as 3.14?

a. After 11 days, approximately 91.91% of the animals will be affected.

b. It will take around 20.83 days for 90% of the animals to become infected. The answer, rounded to two decimal places, is 20.83 days.

A logarithm is defined as the number of powers to which a number must be increased in order to obtain some other numbers. It is the simplest way to express enormous numbers. A logarithm has several key features that demonstrate that logarithm multiplication and division can also be represented in the form of logarithm addition and subtraction.

A) To find the percentage of animals infected after 11 days, we simply substitute t = 11 into the given equation for V(t):

V(11) = 100/ [tex](1 + 99e^{(-0.186*11)})[/tex]

Using a calculator, we get:

V(11) ≈ 91.91%

Therefore, about 91.91% of the animals will be infected after 11 days.

B) To find the time it takes until exactly 90% of the animals are infected, we need to solve the equation V(t) = 90 for t.

Substituting V(t) into the equation, we get:

90 = 100/ [tex](1 + 99e^{(-0.186t)})[/tex]

Multiplying both sides by [tex](1 + 99e^{(-0.186t)})[/tex], we get:

[tex]90 + 90e^{(-0.186t)} = 100[/tex]

Simplifying, we get:

[tex]e^{(-0.186t)} = 1/9[/tex]

Taking the natural logarithm of both sides, we get:

-0.186t = ln(1/9)

Solving for t, we get:

t ≈ 20.83 days

Therefore, about 20.83 days will elapse until exactly 90% of the animals are infected. Rounded to 2 decimal places, the answer is 20.83 days.

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If the cube with 2.00 m wide and 2.00 m long and 2.00 m high has a weight of 900.00 n, then the cube exerts a pressure of 225 N/m².

To calculate the pressure exerted by the cube, follow these steps:

Step 1: To calculate the pressure exerted by the cube,  you need to consider its weight and the area over which it is exerting the force. The cube has a weight of 900 N and dimensions of 2.00 m x 2.00 m x 2.00 m.

Step 2: To find the pressure, we will use the formula:

Pressure (P) = Force (F) / Area (A)

Step 3: In this case, the force is the weight of the cube (900 N), and the area is the base of the cube (2.00 m x 2.00 m).

A = 2.00 m * 2.00 m = 4.00 m²

Step 4: Now, you can calculate the pressure:

P = 900 N / 4.00 m² = 225 N/m²

So, the cube exerts a pressure of 225 N/m².

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If Rolinda’s first five Spanish test scores are 85, 85, 60, 62, and 59.

a. The mean is 70.2, media is 62.

b. The mean is the measure that best  support Rolinda’s claim

c. Rolinda's claim misleading since the two high scores of 85 inflate her mean score of 70.2.

a. Mean

Mean = (85 + 85 + 60 + 62 + 59) / 5

Mean = 70.2

We must first rank the scores from lowest to highest in order to find the median:

59, 60, 62, 85, 85

So.

Median score is 62

We search for the score that shows up most frequently to determine the mode. Two scores 85 appear twice in this instance while the other scores only appear once. Based on this the group of scores does not have a special mode.

b. The mean is the measure that best  support Rolinda’s claim based on the fact that the mean includes all the scores and is influenced by both the high and low scores.

c. Rolinda's claim misleading since the two high scores of 85 inflate her mean score of 70.2.

Therefore the mean is 70.2.

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The probability that all 5 persons take more than 1 minute is 0.1888.

The probability that one person takes more than 1 minute to complete a phone call is given by:

[tex]P(X > 1) = e^(-1/3)[/tex]

So, the probability that all 5 persons take more than 1 minute is:

P(X1 > 1 and X2 > 1 and X3 > 1 and X4 > 1 and X5 > 1) = P(X > 1)^5

Substituting the value of P(X > 1), we get:

P(X1 > 1 and X2 > 1 and X3 > 1 and X4 > 1 and X5 > 1) = (e^(-1/3))^5

Simplifying, we get:

P(X1 > 1 and X2 > 1 and X3 > 1 and X4 > 1 and X5 > 1) = e^(-5/3)

Using a calculator, we get:

P(X1 > 1 and X2 > 1 and X3 > 1 and X4 > 1 and X5 > 1) ≈ 0.03577

Therefore, the answer is c. 0.03577.
To answer your question, we will use the exponential distribution and its properties.

Given the expected value μ = 3 minutes, we can find the parameter λ by using the formula μ = 1/λ. Thus, λ = 1/3 per minute.

Now, we need to find the probability that a single person takes more than 1 minute to complete a phone call. This is equivalent to finding the probability P(X > 1). We can use the cumulative distribution function (CDF) of the exponential distribution for this purpose: P(X > x) = 1 - P(X ≤ x) = [tex]1 - (1 - e^(-λx)).[/tex]

Plugging in λ = 1/3 and x = 1, we get:

P(X > 1) = 1 - (1 - e^(-1/3)) ≈ 0.71653.

Since the phone calls are independent events, the probability that all 5 persons take more than 1 minute is:

P(All > 1) = [tex](0.71653)^5[/tex] ≈ 0.1888.

So, the correct answer is option b. 0.1888.

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The volume of the rectangular prism is 12 3/4 cubic feet. Option C

The formula that is used to calculate the volume of a rectangular prism is expressed as;

V = lwh

Such that the parameters are;

From the information given, we have;

Length = 2 ft

Width = 3/2 feet

height = 17/4 feet

Substitute the values

Volume = 2 × 3/2 × 17/4

volume = 102/8

Volume = 12 3/4 cubic feet

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Answer:

  1.33×10²⁷ kg

Step-by-step explanation:

You want the difference in masses of Jupiter and Saturn in standard form.

The difference of numbers in scientific notation is best found by expressing each number with the same exponent of 10. Here, that difference is ...

  [tex]1.898\times10^{27}-5.68\times10^{26}\\\\=1.898\times10^{27}-0.568\times10^{27}\\\\=(1.898-0.568)\times10^{27}=\boxed{1.33\times10^{27}}[/tex]

__

Additional comment

In the US, "standard form" is the "ordinary number". It will have a total of 28 digits.

1,330,000,000,000,000,000,000,000,000

Your calculator can find the difference for you, and express it in whatever form you want.

Answer:

This is false as you can have triangular sequences.

The second derivative test of the function is solved and the local maximum point of the function is at x = -1/2

Given data ,

Let the function be represented as A

Now , the value of A is

f ( x ) = 2x³ + 3x² - 36x + 5

Now , the first derivative of f(x) to obtain f'(x) is

f'(x) = 6x² + 6x - 36

And , the second derivative of f(x) by differentiating f'(x) with respect to x is

f''(x) = 12x + 6

Now , Set f''(x) = 0 and solve for x to find the critical points.

12x + 6 = 0

12x = -6

x = -6/12

x = -1/2

For x < -1/2: Since f''(x) = 12x + 6, and x < -1/2, the value of f''(x) will be negative, indicating that the function is concave down in this interval, and there is no local maximum point.

For x > -1/2: Since f''(x) = 12x + 6, and x > -1/2, the value of f''(x) will be positive, indicating that the function is concave up in this interval, and there may be a local maximum point.

And , If f'(x) is continuous at x = -1/2, then there must be a local maximum point at x = -1/2 since f''(x) changes sign at x = -1/2. We may verify the value of f'(x) at x = -1/2 to see if f'(x) is continuous at x = -1/2.

f'(-1/2) = 6(-1/2)² + 6(-1/2) - 36 = 3 - 3 - 36 = -36

Since f'(-1/2) = -36 is a finite function, we may infer that f'(x) is continuous at x = -1/2 and that x = -1/2 is the location of the local maximum

Hence , the local maximum is at x = -1/2

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The absolute extrema of the function g(x) = 3x²/(x - 2) on the closed interval [-2, 1] are

a) Minimum: (1, -9)

b) Maximum: (4, 24)

To find the absolute extrema of the function g(x) = 3x²/(x - 2) on the closed interval [-2, 1], we need to evaluate the function at the critical points and endpoints of the interval.

First, we need to find the critical points of the function, which occur when the derivative of g(x) is equal to zero or undefined. We have

g(x) = 3x²/(x - 2)

g'(x) = (6x(x - 2) - 3x²)/ (x - 2)²

g'(x) = (3x(x - 4))/ (x - 2)²

Setting g'(x) equal to zero, we get

3x(x - 4) = 0

x = 0 or x = 4

Note that x = 2 is not in the domain of the function, so it is not a critical point.

Next, we need to evaluate the function at the critical points and endpoints of the interval. We have

g(-2) = 12

g(0) = 0

g(1) = -9

g(4) = 24

Therefore, the absolute maximum of the function on the interval is g(4) = 24, which occurs at x = 4. The absolute minimum of the function on the interval is g(1) = -9, which occurs at x = 1.

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The given question is incomplete, the complete question is:

Find the absolute extrema of the function on the closed interval.g(x)=3x²/x-2, [-2,1].

The line through the point (2,0,0) and with direction vector (-1, a, 1) intersects the paraboloid z = x² + y² in only a single point for a = 0 or a = √(2).

To learn more about here:

To find the intersection points, we can substitute the equation of the line into the equation of the paraboloid:

z = x² + y²
z = (2-t)² + a*t²
x = 2-t
y = a*t
where t is the parameter for the line.

Substituting x and y into the equation of the paraboloid gives:
z = (2-t)² + a^2*t²

To find the values of a where the line intersects the paraboloid in only a single point, we need to find the values of a where this equation has exactly one solution. This occurs when the discriminant of the quadratic equation in t is zero:

a²*(2-a²) = 0
a = 0 or a = √(2)

Therefore, the line through the point (2,0,0) and with direction vector (-1, a, 1) intersects the paraboloid z = x² + y² in only a single point for a = 0 or a = √(2).

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The answer is 10, which corresponds to option (E).

To solve it, let's first analyze the given terms and find a pattern:

1. DOG = 29
2. BAG = 13
3. FEE = 19

Now, let's convert each letter to its corresponding position in the alphabet:
- D = 4, O = 15, G = 7
- B = 2, A = 1, G = 7
- F = 6, E = 5, E = 5

Next, let's look for a pattern in the sums:
1. 4 + 15 + 7 = 26 → 26 + 3 = 29
2. 2 + 1 + 7 = 10 → 10 + 3 = 13
3. 6 + 5 + 5 = 16 → 16 + 3 = 19

It appears that after summing the positions of each letter, we add 3 to get the final result.
Now, let's find the value for DAB:
- D = 4, A = 1, B = 2
- 4 + 1 + 2 = 7 → 7 + 3 = 10
So, the answer is 10, which corresponds to option (E).

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The area of the region R = {(x, y) | x² + y² ≤ 25, x ≥ 4} using polar coordinates is 7π square units.

To calculate the area, first, we need to convert the given equations into polar coordinates. The equation x² + y² ≤ 25 becomes r² ≤ 25, which simplifies to 0 ≤ r ≤ 5. The equation x ≥ 4 can be written as r*cos(θ) ≥ 4. Solving for θ, we get 0 ≤ θ ≤ 2π/3 and 4π/3 ≤ θ ≤ 2π.

Now, use the polar area formula: A = 0.5 * ∫(r² dθ). Integrate r²/2 from 0 to 2π/3 and from 4π/3 to 2π, then multiply by the limits' difference. Finally, add the two areas to find the total area of the region, which is 7π square units.

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A tree diagram for this scenario would have three branches for screen sizes (10, 12, 15 inches) and then four branches for each of those screen sizes representing the hard drive sizes (50, 100, 150, 200 GB).
b. - A ∩ B: {(10, 50), (10, 100), (12, 50), (12, 100)}

c-- B and C are disjoint events, as they have no common outcomes.

a. A tree diagram for this scenario would have three branches for screen sizes (10, 12, 15 inches) and then four branches for each of those screen sizes representing the hard drive sizes (50, 100, 150, 200 GB).

b.
- A (screen size of 12 inches or smaller): {(10, 50), (10, 100), (10, 150), (10, 200), (12, 50), (12, 100), (12, 150), (12, 200)}
- B (hard drive size of at most 100 GB): {(10, 50), (10, 100), (12, 50), (12, 100), (15, 50), (15, 100)}

- AC: {(10, 150), (10, 200), (12, 150), (12, 200)}
- A ∪ B: All outcomes except for {(15, 150), (15, 200)}
- A ∩ B: {(10, 50), (10, 100), (12, 50), (12, 100)}

c.
- C (order for a laptop with a 200 GB hard drive): {(10, 200), (12, 200), (15, 200)}

- A and C are not disjoint events, as they share common outcomes {(10, 200), (12, 200)}.
- B and C are disjoint events, as they have no common outcomes.

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Answer: 400 miles

Step-by-step explanation: every hour, his distance decreases by 50 miles. After 4 hours, his distance is 200 miles, so 50 miles times 4 hours = 200 miles+the original 200 miles =400 miles.

Answer:

$6133.19

Step-by-step explanation:

We can use the formula for compound interest to find the amount of money in Wazin's college account in 2026:

A = P(1 + r/n)^(nt)

where A is the amount of money in the account, P is the principal (initial investment), r is the interest rate (as a decimal), n is the number of times the interest is compounded per year, and t is the time (in years).

In this case, P = $1500, r = 0.07, n = 4 (since the interest is compounded quarterly), and t = 20 (since 2026 is 20 years after 2006). Substituting these values, we get:

A = 1500(1 + 0.07/4)^(4*20) = $6133.19

Therefore, Wazin's college account will have approximately $6133.19 in 2026.

Hope this helps!

Answer:

Step-by-step explanation:

Principal amount, P= $1500 Rate of interest, r = 7%

I may or may not be lying. >:^P

Using the standard normal distribution table, we can look up the probability corresponding to a z-score of 1.89 and subtract from it the probability corresponding to a z-score of -1.54, as follows:

P(-1.54 < z < 1.89) = P(z < 1.89) - P(z < -1.54)

Looking up these probabilities in the standard normal distribution table, we find:

P(z < 1.89) = 0.9706

P(z < -1.54) = 0.0621

Substituting these values into the formula, we get:

P(-1.54 < z < 1.89) = 0.9706 - 0.0621 = 0.9085

Therefore, the probability that z is between -1.54 and 1.89 is approximately 0.9085, or 90.85% (rounded to two decimal places).

*IG: whis.sama_ent*

The equivalent rates of the cashier are  4 customers/20 minutes and 0.2 customers/minutes

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

Served four customers in 20 minutes

This means that

Customers = 4

Time = 20 minutes

So, the rate is

Rate = customers/Time

Substitute the known values in the above equation, so, we have the following representation

Rate = 4 customers/20 minutes

When converted to equivalent rates, we have

Rate = 0.2 customers/minutes

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The point estimate of the population mean is 18.4. Therefore, the correct answer is (d) is 18.4

When conducting a statistical study, it is important to have a good understanding of the population in question. In such cases, a sample of the population can be taken to infer information about the population.

One of the key parameters that is of interest is the population mean. The population mean represents the average value of a particular characteristic in the entire population. However, since it is usually not possible to collect data from the entire population, we use the sample mean as an estimate of the population mean.

In the given scenario, a simple random sample of a population was taken, and the following data was collected: 13, 17, 18, 21, 23. The point estimate of the population mean can be calculated by taking the mean of the sample.

The sample mean is calculated as follows:

(13 + 17 + 18 + 21 + 23) / 5 = 92 / 5 = 18.4

Therefore, the point estimate of the population mean is 18.4, option (d).

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In this case, the p-value of .021 is greater than the alpha level of .01. Therefore, we fail to reject the null hypothesis. This means that we do not have enough evidence to conclude that the alternative hypothesis is true. The correct answer is D.

When conducting hypothesis testing, the a priori significance level (alpha) is set before the data is analyzed. This level is the threshold for determining whether the test statistic is significant or not. In this case, the alpha is set at .01, meaning that the probability of rejecting the null hypothesis when it is true is 1 in 100.

The test statistic is the calculated value that is used to determine whether the null hypothesis should be rejected or not. In this case, the test statistic has a p-value of .021. The p-value is the probability of obtaining a test statistic as extreme as or more extreme than the observed statistic, assuming the null hypothesis is true.

In other words, it tells us how likely it is that the observed data occurred by chance alone. To determine what to do next, we compare the p-value to the alpha level. If the p-value is less than or equal to the alpha level, then we reject the null hypothesis. If the p-value is greater than the alpha level, then we fail to reject the null hypothesis.

In this case, the p-value of .021 is greater than the alpha level of .01. Therefore, we fail to reject the null hypothesis. This means that we do not have enough evidence to conclude that the alternative hypothesis is true. It is important to note that failing to reject the null hypothesis does not mean that the null hypothesis is true, only that we do not have enough evidence to reject it.

There is no need to calculate the SEM (standard error of the mean) or accept the alternative hypothesis in this scenario. The correct answer is D, fail to reject the null hypothesis.

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By answering the presented question, we may conclude that The derivative of the function u(t) is therefore [tex](10t^4 - 14t^2)j.[/tex]

Mathematics is concerned with numbers and their variations, equations and related structures, shapes and their placements, and locations where they may be found.

The term "function" refers to the link between a set of inputs, each of which has an associated output. A function is a relationship between inputs and outputs that produces a single, distinct result for each input.

Each function is given a domain and a codomain, or scope. The letter f is frequently used to represent functions (x). An x is used as the input. The four basic kinds of functions offered are on functions, one-to-one functions, many-to-one functions, within functions, and on functions.

The function you supplied is as follows:

To calculate the derivative of this function, we must take the derivative of each component with respect to t independently.

The product rule of differentiation may be used to find the derivative of the first component [tex]t, 2t^3 (t^2-7).[/tex]

Let f(t) = [tex]2t^3[/tex]and g(t) = [tex]t^2[/tex] - 7. Then, using the product rule, we obtain:

Because k is a constant, the derivative of the second component, -5k, is simply zero.

As a result, the derivative of the function u(t) with respect to t is as follows:

The derivative of the function u(t) is therefore[tex](10t^4 - 14t^2)j.[/tex]

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The length of the equiangular spiral r =  [tex]e^{\theta}[/tex]  for 0 ≤ θ ≤ 2/10 pi is approximately 1.8315.

To find the length of the equiangular spiral r =  [tex]e^{\theta}[/tex]  for 0 ≤ theta ≤ 2/10 pi, we use the formula for the arc length of a polar curve: L = ∫√(r² + (dr/dθ)²) dθ.

For r = [tex]e^{\theta}[/tex] the derivative dr/dθ =  [tex]e^{\theta}[/tex] . Now, we can find the arc length L:

L = ∫(from 0 to 2/10 pi) √(( [tex]e^{\theta}[/tex] )² + ( [tex]e^{\theta}[/tex] )²) dθ.  

By factoring out [tex]e^{2\theta}[/tex], we get:

L = ∫(from 0 to 2/10 pi)  [tex]e^{\theta}[/tex]  √(1 + 1) dθ.

Next, integrate:

L =  [tex]e^{\theta}[/tex] (√2) | from 0 to 2/10 pi.

Evaluating the integral:

L = (√2)([tex]e^\frac{2}{10} ^{\pi}[/tex] - e⁰).

L ≈ 1.8315.

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1. The equations for Sand, Clay, and Humus, we get: x = (12 - 3y)/4, x = (12 - 6y)/5, x = (10 - 12y)/5 2. The feasible region is the shaded region above the line Sand = 3 and to the left of the line Clay = 2.

What is linear programming?

Linear programming is a mathematical method used to optimize a linear objective function subject to linear constraints.

1. Formulating the problem as a linear programming:

Let x and y be the number of cubic feet of compost and topsoil, respectively, used to make one bag of potting soil.

We want to minimize the cost of the potting soil, which is given by:

Cost = 0.12x + 0.2y

We want to ensure that each bag of potting soil contains at least 12 pounds of sand, 12 pounds of clay, and 10 pounds of humus. The amount of each ingredient in one bag of potting soil can be calculated as follows:

Sand = 4x + 3y

Clay = 5x + 6y

Humus = 5x + 12y

We can now formulate the constraints as follows:

Sand ≥ 12

Clay ≥ 12

Humus ≥ 10

Solving for x and y in the equations for Sand, Clay, and Humus, we get:

x = (12 - 3y)/4

x = (12 - 6y)/5

x = (10 - 12y)/5

We also have the non-negativity constraints:

x ≥ 0

y ≥ 0

2. Plotting the constraints and showing the feasible region:

We can graph the constraints by plotting the equations for Sand, Clay, and Humus, and shading the region that satisfies all the constraints. The resulting feasible region is shown below:

The feasible region is the shaded region above the line Sand = 3 and to the left of the line Clay = 2.

3. Identifying the optimal solution:

We can find the optimal solution by finding the point in the feasible region that minimizes the cost of the potting soil. This point occurs where the cost function is minimized.

Cost = 0.12x + 0.2y

Substituting x = (12 - 3y)/4 and x = (12 - 6y)/5, we get:

Cost = 0.12[(12 - 3y)/4] + 0.2y

Cost = 0.12[(12 - 6y)/5] + 0.2y

Simplifying, we get:

Cost = 0.6 - 0.09y

Cost = 0.72 - 0.044y

We can now find the minimum value of Cost by setting its derivative to zero:

dCost/dy = -0.09 + 0.044 = 0

Solving for y, we get:

y = 2

Substituting y = 2 into x = (12 - 3y)/4 and x = (12 - 6y)/5, we get:

x = 3/4

x = 6/5

Therefore, the optimal solution occurs at x = 3/4 and y = 2, and the minimum cost of the potting soil is:

Cost = 0.12x + 0.2y = 0.12(3/4) + 0.2(2) = 0.39 dollars.

4. Interpreting the optimal solution:

The optimal solution indicates that the Queen City Nursery should use 3/4 cubic feet of compost and 2 cubic feet of topsoil to make one bag of potting soil, which will cost 39 cents. This solution satisfies all the constraints and minimizes the cost of the potting soil. The optimal solution also indicates that the potting soil should contain 3 cubic feet of sand, 12 cubic feet of clay, and 11 cubic feet of humus, which satisfies the minimum requirements for each ingredient.

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Answer:

3x12.50 no need equation

Answer:

3x12.50 no need equation

Marginal cost (MC) [tex]= dTC/dQ = d(2L + 64)/dQ = 2(3/2)L^{-2/3} = 3L^{-2/3}[/tex]

We must first determine the total cost (TC), variable cost (VC), and fixed cost (FC) for the given production function Q =[tex]4K^{1/3}L^{2/3}[/tex], where K is fixed at 8 units, before sketching the cost curves. The sum of the variable cost and the fixed cost is the total cost:

TC = VC + FC

The variable cost is the cost of variable inputs, which in this case is labor (L), and is given by the equation:

VC = wL

where w is the wage rate. The fixed cost is the cost of fixed inputs, which in this case is capital (K), and is given by the equation:

FC = rK

where r is the rental rate of capital.

We must also calculate the average total cost (ATC), average variable cost (AVC), average fixed cost (AFC), and marginal cost (MC) in order to calculate the cost curves. The equations that follow describe these:

ATC = TC/Q

AVC = VC/Q

AFC = FC/Q

MC = dTC/dQ

Now, let's calculate the cost curves.

Since K is fixed at 8 units, the production function becomes:

Q = [tex]4(8)^{1/3}L^{2/3}[/tex]

Simplifying this equation, we get:

Q = [tex]16L^{2/3}[/tex]

Taking the derivative of the production function with respect to L, we get the marginal product of labor (MPL):

MPL = dQ/dL = (32/3)L^-1/3

Now, we can calculate the cost curves:

Variable cost (VC) = wL = 2L

Fixed cost (FC) = rK = 8(8) = 64

Total cost (TC) = VC + FC = 2L + 64

Average variable cost (AVC) = VC/Q =[tex](2L)/16L^{2/3} = 2L^{1/3}/16[/tex]

Average fixed cost (AFC) = FC/Q = [tex]64/16L^{2/3} = 4/L^{2/3}[/tex]

Average total cost (ATC) = TC/Q =[tex](2L + 64)/16L^{2/3} = (2L^{1/3} + 64/L^{2/3})/16[/tex]

Marginal cost (MC) [tex]= dTC/dQ = d(2L + 64)/dQ = 2(3/2)L^{-2/3} = 3L^{-2/3}[/tex]

Now, Cost curves

The quantity of output (Q) is shown on the x-axis, and the cost is shown on the y-axis. Each line on the vertical axis is scaled to be a multiple of 10 on the scale.

The variable cost curve (VC) is a straight line with a slope of 2 that runs through the origin. The fixed cost curve (FC) is a 64-degree horizontal line.

The sum of the VC and FC curves is the total cost curve (TC). It is a straight line that goes through the point (0, 64) and has a slope of 2.

U-shaped is the average variable cost curve (AVC). At its smallest point, it comes to a stop at the MC curve.

As output rises, the average fixed cost curve (AFC) slopes downward.

U-shaped is the average total cost curve (ATC). At its smallest point, it comes to a stop at the MC curve.

The minimum points of the AVC and ATC curves are intersected by the marginal cost curve (MC), which has a slope that is downward.

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The sum of 7/10 and 2/5 is 29/50.

Answer: 11/10 is the correct answer to this question.

Step-by-step explanation:

First, we take 7/10 and 2/5. The L.C.M of denominators is equal to 10. As the denominator in 7/10 is already 10 we don't change it but as the denominator in 2/5 is 5 we multiply the numerator and denominator with 2 in order to equalize both. Hence we get 7/10 and 4/10. On adding both the numerators we get 11/10.

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a. Using poisson distribution The average time a car is in the system, considering waiting in line and receiving service, is 12 minutes.

b. The average number of cars in the system, considering both waiting in line and receiving service, is 10.8 cars.

c. The average number of cars waiting to receive service is 10 cars.

a. The average time a car is in the system can be calculated as the sum of the average time spent waiting in line and the average time spent receiving service. Let's calculate each of these separately:

Average time spent waiting in line: Since arrivals follow a Poisson distribution with a rate of 4 every 10 minutes, the interarrival time (time between consecutive arrivals) follows an exponential distribution with parameter λ = 4/10 = 0.4.

The average interarrival time is 1/λ = 2.5 minutes. By Little's law, the average number of cars waiting in line is equal to the product of the arrival rate and the average time spent waiting, which is 4(2.5) = 10 minutes.

Average time spent receiving service: Since service times are exponentially distributed with a mean of 2 minutes, the average time spent receiving service is also 2 minutes.

Therefore, the average time a car is in the system is 10 + 2 = 12 minutes.

b. The average number of cars in the system can be calculated as the sum of the average number of cars waiting in line and the average number of cars receiving service. Since service times are exponentially distributed and arrivals follow a Poisson distribution, the system can be modeled as an M/M/1 queue.

The average number of cars waiting in line can be calculated using Little's law, which gives us 4(2.5) = 10 cars. The average number of cars receiving service can be calculated as the ratio of the average service time to the average interarrival time, which is 2/2.5 = 0.8 cars. Therefore, the average number of cars in the system is 10 + 0.8 = 10.8 cars.

c. The average number of cars waiting to receive service can be calculated as the difference between the average number of cars in the system and the average number of cars receiving service. Therefore, the average number of cars waiting to receive service is 10.8 - 0.8 = 10 cars.

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