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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.]

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## 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: $C'(x) = 0 + 40 - 0.002x \\ C'(x) = 40 - 0.002x$

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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