Chapter 11: Chapter 11

Problem 10

The cost of producing x teddy bears per day at the Cuddly Companion Co. is calculated by their marketing staff to be given by the formula $$ C(x)=100+40 x-0.001 x^{2} $$ a. Find the marginal cost function and use it to estimate how fast the cost is going up at a production level of 100 teddy bears. Compare this with the exact cost of producing the 101st teddy bear. HINT [See Example 1.] b. Find the average cost function $\bar{C}$, and evaluate $\bar{C}(100)$. What does the answer tell you? HINT [See Example 4.]

Short Answer

Expert verified

The marginal cost of producing 100 teddy bears is approximately \(39.80, the exact cost of producing the 101st teddy bear is $40.01, and the average cost of producing 100 teddy bears is $38.20. The average cost tells us the cost per teddy bear at a production level of 100 teddy bears.

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Step 1: Find the marginal cost function

The marginal cost function represents the rate at which the total cost is changing with respect to the number of teddy bears produced. To find the marginal cost function, we need to find the first derivative of the given cost function with respect to x. The given cost function: $C(x) = 100 + 40x - 0.001x^2$ Differentiating with respect to x, we get: \[C'(x) = \frac{d}{dx}(100 + 40x - 0.001x^2)\]

Step 2: Differentiate the cost function

To differentiate the cost function $C(x)$, first, we find the derivative of each term individually. \[ \frac{d}{dx}(100) = 0 \\ \frac{d}{dx}(40x) = 40 \\ \frac{d}{dx}(- 0.001x^2) = -0.002x \\ \] Thus, the marginal cost function is: \[ C'(x) = 0 + 40 - 0.002x \\ C'(x) = 40 - 0.002x \]

Step 3: Estimate the cost at 100 teddy bears

Now that we have the marginal cost function, we can use it to estimate the cost of producing 100 teddy bears. Evaluate the marginal cost function at x = 100: \[ C'(100) = 40 - 0.002(100) \\ C'(100) = 40 - 0.2 \\ C'(100) = 39.8 \] The marginal cost of producing 100 teddy bears is approximately $39.80.

Step 4: Calculate the exact cost of producing the 101st teddy bear

To find the exact cost of producing the 101st teddy bear, we need to evaluate the cost function at $x = 101$ and subtract the cost at $x = 100$. \[ C(101) = 100 + 40(101) - 0.001(101^2) \\ C(101) ≈ 3860.01 \] \[ C(100) = 100 + 40(100) - 0.001(100^2) \\ C(100) ≈ 3820.00 \] Exact cost of producing the 101st teddy bear: \[ C(101) - C(100) ≈ 3860.01 - 3820.00 \\ C(101) - C(100) ≈ 40.01 \]

Step 5: Find the average cost function

The average cost function $\bar{C}(x)$ is given by the ratio of the total cost function $C(x)$ to the number of teddy bears produced (x): \[ \bar{C}(x) = \frac{C(x)}{x} \] Substitute the given cost function: \[ \bar{C}(x) = \frac{100 + 40x - 0.001x^2}{x} \]

Step 6: Evaluate the average cost function at 100 teddy bears

Now we shall evaluate the average cost function $\bar{C}(x)$ at x = 100. \[ \bar{C}(100) = \frac{100 + 40(100) - 0.001(100^2)}{100} \\ \bar{C}(100) = \frac{3820}{100} \\ \bar{C}(100) = 38.20 \] From our calculations, we found that the marginal cost of producing 100 teddy bears is approximately $39.80, the exact cost of producing the 101st teddy bear is $40.01, and the average cost of producing 100 teddy bears is $38.20. The average cost tells us the cost per teddy bear at a production level of 100 teddy bears.

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Chapter 11: Chapter 11

Short Answer

Step by step solution

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Step 1: Find the marginal cost function

Step 2: Differentiate the cost function

Step 3: Estimate the cost at 100 teddy bears

Step 4: Calculate the exact cost of producing the 101st teddy bear

Step 5: Find the average cost function

Step 6: Evaluate the average cost function at 100 teddy bears

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Chapter 0

Chapter 1

Chapter 2

Chapter 3

Chapter 4

Chapter 5

Chapter 6

Chapter 7

Chapter 8

Chapter 9

Chapter 10

Chapter 12

Chapter 13

Chapter 14

Chapter 15

Chapter 16

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