Joseph Chen
06/11/2023 · High School
[0/11.11 Points] DETAILS A car wash firm calculates that its daily profit (in dollars) depends on the number \( n \) of workers it employs according to the formula \[ P=-600 n+25 n^{2}-0.005 n^{4} \text {. } \] Calculate the marginal product at an employment level of 50 workers. HINT [See Example 3.] \( \$ \$ \)
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To calculate the marginal product at an employment level of 50 workers, we need to find the derivative of the profit function with respect to the number of workers \( n \) and then evaluate it at \( n = 50 \).
Given the profit function:
\[ P = -600n + 25n^2 - 0.005n^4 \]
The marginal product is the derivative of the profit function with respect to the number of workers \( n \). Let's find the derivative of the profit function.
Evaluate the derivative by following steps:
- step0: Evaluate the derivative:
\(\frac{d}{dn}\left(-600n+25n^{2}-0.005n^{4}\right)\)
- step1: Use differentiation rules:
\(-\frac{d}{dn}\left(600n\right)+\frac{d}{dn}\left(25n^{2}\right)-\frac{d}{dn}\left(0.005n^{4}\right)\)
- step2: Calculate:
\(-600+50n-0.02n^{3}\)
The derivative of the profit function with respect to the number of workers \( n \) is:
\[ \frac{dP}{dn} = -600 + 50n - 0.02n^3 \]
Now, we need to evaluate this derivative at \( n = 50 \) to find the marginal product at an employment level of 50 workers. Let's substitute \( n = 50 \) into the derivative expression and calculate the result.
Evaluate the expression by following steps:
- step0: Evaluate:
\(-600+50n-0.02n^{3}\)
- step1: Substitute:
\(-600+50\times 50-0.02\times 50^{3}\)
- step2: Multiply the numbers:
\(-600+2500-0.02\times 50^{3}\)
- step3: Multiply the numbers:
\(-600+2500-2500\)
- step4: Calculate:
\(-600\)
The marginal product at an employment level of 50 workers is $-600.
Therefore, the marginal product at an employment level of 50 workers is $-600.
Quick Answer
The marginal product at 50 workers is $-600.
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