Question

an analysis of the daily output of a factory assembly line shows that about $30t + t^2 - \frac{1}{48}t^3$ units are produced after $t$ hours of work, $0 \leq t \leq 8$. what is the rate of production (in units per hour) when $t = 4$? at $t = 4$, the rate of production is \\(\square\\) units per hour.

Explanation:

Step1: Identify the production function

The production function is \( P(t) = 30t + t^2 - \frac{1}{48}t^3 \). To find the rate of production, we need the derivative of \( P(t) \) with respect to \( t \), which represents the marginal production (rate of production).

Step2: Find the derivative of \( P(t) \)

Using the power rule for differentiation (\( \frac{d}{dt} t^n = nt^{n - 1} \)):

• The derivative of \( 30t \) is \( 30 \).
• The derivative of \( t^2 \) is \( 2t \).
• The derivative of \( -\frac{1}{48}t^3 \) is \( -\frac{1}{48} \times 3t^2 = -\frac{t^2}{16} \).

So, the rate of production function \( r(t) = P'(t) = 30 + 2t - \frac{t^2}{16} \).

Step3: Evaluate \( r(t) \) at \( t = 4 \)

Substitute \( t = 4 \) into \( r(t) \):
\[

$$\begin{align*} r(4) &= 30 + 2(4) - \frac{(4)^2}{16}\\ &= 30 + 8 - \frac{16}{16}\\ &= 30 + 8 - 1\\ &= 37 \end{align*}$$

\]

Answer:

37