Questions: The weekly cost in dollars, C, to manufacture noise cancelling headphones depends on the number of quality control officers, q, according to the model C(q)=55q+135q^(1/5). In order to produce k noise cancelling headphones a week, they need φ(k)=(300/k)+0.2k quality control officers. a. What are their weekly costs when they produce 60 noise cancelling headphones a week? Round your final answer to the nearest whole number and include units with your answers. b. At what rate is the company's weekly cost changing with respect to the number of noise cancelling headphones they produce, when they make 60 noise cancelling headphones a week? Round your final answer to 2 decimal places and include units with your answer.

Solution Steps

To solve the given problem, we need to follow these steps:

1. Calculate the number of quality control officers required to produce 60 noise-cancelling headphones using the function \(\varphi(k)\).
2. Use the calculated number of quality control officers to determine the weekly cost using the function \(C(q)\).
3. Round the final cost to the nearest whole number and include units.

1. Find the derivative of the cost function \(C(q)\) with respect to \(q\).
2. Calculate the rate of change of the company's weekly cost with respect to the number of noise-cancelling headphones produced when \(k = 60\).
3. Round the final answer to 2 decimal places and include units.

To determine the number of quality control officers required to produce \( k = 60 \) noise-cancelling headphones, we use the function:

\[ \varphi(k) = \frac{300}{k} + 0.2k \]

Substituting \( k = 60 \):

\[ \varphi(60) = \frac{300}{60} + 0.2 \times 60 = 5 + 12 = 17 \]

Thus, the number of quality control officers needed is \( q = 17 \).

Next, we calculate the weekly cost using the cost function:

\[ C(q) = 55q + 135q^{1/5} \]

Substituting \( q = 17 \):

\[ C(17) = 55 \times 17 + 135 \times 17^{1/5} \approx 1172.9159 \]

Rounding to the nearest whole number, the weekly cost is:

\[ \text{Weekly Cost} = 1173 \]

To find the rate at which the company's weekly cost is changing with respect to the number of noise-cancelling headphones produced, we first compute the derivative of the cost function:

\[ \frac{dC}{dq} = 55 + 27q^{-4/5} \]

Evaluating this derivative at \( q = 17 \):

\[ \frac{dC}{dq}\bigg|_{q=17} \approx 57.7990 \]

Rounding to two decimal places, the rate of change of the weekly cost is:

\[ \text{Rate of Change} \approx 57.80 \]

Final Answer