Question

a company finds that the cost, in dollars, of producing ( t ) handbags is given by ( c(t) = 440 + 30t + 17t^2 ). find the rate at which average cost, ( \frac{c(t)}{t} ), is changing when 150 handbags have been produced. (round to the nearest cent.) the average cost is? by? when 150 handbags are produced.

Explanation:

Step1: Define Average Cost Function

The average cost function \(\bar{C}(t)\) is given by \(\bar{C}(t)=\frac{C(t)}{t}\). Given \(C(t) = 440+30t + 17t^{2}\), so \(\bar{C}(t)=\frac{440 + 30t+17t^{2}}{t}=\frac{440}{t}+30 + 17t\).

Step2: Find Derivative of Average Cost

To find the rate of change of average cost, we take the derivative of \(\bar{C}(t)\) with respect to \(t\). Using the power rule, the derivative of \(\frac{440}{t}=440t^{- 1}\) is \(- 440t^{-2}=-\frac{440}{t^{2}}\), the derivative of \(30\) is \(0\), and the derivative of \(17t\) is \(17\). So \(\bar{C}'(t)=-\frac{440}{t^{2}}+17\).

Step3: Evaluate at \(t = 150\)

Substitute \(t = 150\) into \(\bar{C}'(t)\):
\[

$$\begin{align*} \bar{C}'(150)&=-\frac{440}{150^{2}}+17\\ &=-\frac{440}{22500}+17\\ &\approx - 0.019555\cdots+17\\ &\approx16.980444\cdots \end{align*}$$

\]
Rounding to the nearest cent (two decimal places), we get approximately \(16.98\) dollars per handbag. Also, the average cost at \(t = 150\) is \(\bar{C}(150)=\frac{440+30\times150 + 17\times150^{2}}{150}=\frac{440 + 4500+17\times22500}{150}=\frac{440+4500 + 382500}{150}=\frac{387440}{150}\approx2582.93\) dollars. But the question first asks for the rate of change of average cost (the derivative value) and then the average cost. Wait, re - reading the problem: "Find the rate at which average cost, \(\frac{C(t)}{t}\), is changing when 150 handbags have been produced. The average cost is? by? when 150 handbags are produced."

Wait, maybe the first blank is the rate of change (derivative) and the second is the average cost.

First, rate of change (derivative at \(t = 150\)):

\(\bar{C}'(t)=-\frac{440}{t^{2}}+17\)

At \(t = 150\):

\(\bar{C}'(150)=17-\frac{440}{150^{2}}=17-\frac{440}{22500}\approx17 - 0.0196=16.9804\approx16.98\) (dollars per handbag, rate of change of average cost)

Average cost \(\bar{C}(150)=\frac{C(150)}{150}=\frac{440 + 30\times150+17\times150^{2}}{150}=\frac{440+4500 + 382500}{150}=\frac{387440}{150}\approx2582.93\)

Answer:

The rate at which average cost is changing (derivative) is approximately \(\$16.98\) per handbag, and the average cost is approximately \(\$2582.93\) when 150 handbags are produced. (If we assume the first blank is the rate of change and the second is the average cost: First blank: \(16.98\), Second blank: \(2582.93\))