**Analysis of Factory Assembly Line Output**An analysis of the daily output of a factory assembly line shows that about $ 50t + \frac{2}{48} t^3 $ units are produced after $ t $ hours of work, $ 0 \leq t \leq 8 $. **Question:**What is the rate of production (in units per hour) when $ t = 4 $?**Solution:**The rate of production is given by the derivative of the output function with respect to time $ t $. The output function is $ 50t + \frac{2}{48} t^3 $. To find the rate of production at $ t = 4 $, we first differentiate the function with respect to $ t $:\[ \frac{d}{dt} \left( 50t + \frac{2}{48}t^3 \right) = 50 + \frac{2}{48} \cdot 3t^2 = 50 + \frac{1}{24} t^2 \]Next, we substitute $ t = 4 $ into the derivative to find the rate of production:\[ \text{Rate of production at } t = 4 = 50 + \frac{1}{24} (4)^2 = 50 + \frac{1}{24} \cdot 16 = 50 + \frac{16}{24} = 50 + \frac{2}{3} = 50.67 \text{ units per hour} \]Thus, the rate of production at $ t = 4 $ is $ \boxed{50.67} $ units per hour.

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Answered: An analysis of the daily output of a…

Calculus: Early Transcendentals

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Chapter1: Functions And Models

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Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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An analysis of the daily output of a factory assembly line shows that about 50t+t²- 1 tunits are produced after t hours of work, 0≤t≤8. What is the rate of production (in units per hour) when t = 4? 48 34 At t= 4, the rate of production is units per hour.