Question

The management of Duff’s restaurant forecast that they will sell 200,000 cases of their chicken wing hot sauce next year. Their annual profit is described by ?(?) = −0.000032?^3 + 6? − 100 where ? is measured in thousands of cases, and ?(?) is measured in thousand dollars. Use differentials to estimate the change in the profit if the sales forecast is decreased by 500 cases, rounding to the nearest dollar. Interpret your answer in a complete sentence using the context of this problem and appropriate units.

Answer 1

given P(x)=-0.000032x³+6x-100
differentiate with respect to x
=>P'(x)=-0.000032(3)x^3-1+6(1)-0
=>P'(x)=-0.000096x²​​​​​​​+6

using diffrentials estimated change in the profit ,ΔP=(P'(x))Δx

? is measured in thousands of cases.sales forecast is decreased by 500 cases.
=>x=200000_/1000=200,Δx=-500_/1000=-0.5

ΔP=(P'(200))Δx
=>ΔP=(-0.000096(200)²​​​​​​​+6)(-0.5)
=>ΔP=(-3.84​​​​​​​+6)(-0.5)
=>ΔP=(2.16)(-0.5)
​​​​​​​=>ΔP=-1.08  thousand dollars
​​​​​​​=>ΔP=-1080  dollars

estimated change in the profit if the sales forecast is decreased by 500 cases is 1080 dollars

profit decreases by 1080 dollars if the sales forecast is decreased by 500 cases