The instantaneous production rates (in widgets per hour) of two factories operating…

MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Integrals · Applications of Integrals

Problem

The instantaneous production rates (in widgets per hour) of two factories operating side-by-side over a 10-hour shift are modeled by

R_1(t)=15\sin\!\left(\frac{\pi t}{10}\right)+20 \quad\text{and}\quad R_2(t)=\frac{t^{2}}{5}+10,

where

t

is measured in hours and

0\le t\le10

. The two rate curves intersect three times within the shift. (a) Verify analytically that

t=0

and

t=10

are intersection points and determine the third intersection time

t=c

0<c<10

, to three decimal places. (b) On the interval

[0,c]

, identify which factory has the higher production rate. Provide analytic or numerical evidence for your claim. (c) Write a single definite integral that represents the total additional widgets produced by the faster factory over the slower one during the first part of the shift,

0\le t\le c

. Do not evaluate the integral in this part. (d) Evaluate the integral from part (c) to the nearest whole widget and interpret your result in the context of factory output. (e) Discuss one managerial implication that could be drawn from the area you calculated in part (d).

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