Question

an analysis of the daily output of a factory assembly line shows that about ( 70t + t^{2}-\frac{1}{12}t^{3} ) units are produced after t hours of work. ( 0leq tleq8 ). what is the rate of production (in units per hour) when ( t = 2 )?
at ( t = 2 ), the rate of production is units per hour.

Explanation:

Step1: Find the derivative of the production function

The production function is \( P(t)=70t + t^{2}-\frac{1}{12}t^{3} \).
Using the power rule \((x^{n})^\prime=nx^{n - 1}\), the derivative \(P^\prime(t)\) is:
\(P^\prime(t)=\frac{d}{dt}(70t)+\frac{d}{dt}(t^{2})-\frac{d}{dt}(\frac{1}{12}t^{3})\)
\(P^\prime(t)=70 + 2t-\frac{1}{12}\times3t^{2}\)
\(P^\prime(t)=70 + 2t-\frac{1}{4}t^{2}\)

Step2: Substitute \(t = 2\) into the derivative

When \(t = 2\), we have:
\(P^\prime(2)=70+2\times2-\frac{1}{4}\times2^{2}\)
First, calculate \(2\times2 = 4\) and \(\frac{1}{4}\times2^{2}=\frac{1}{4}\times4 = 1\)
Then \(P^\prime(2)=70 + 4-1\)
\(P^\prime(2)=73\)

Answer:

73