Estimate the change in profit.

Question

The management off Miller's restaurant forecast that they will sell 200,000 cases of their chicken wing hot sauce next year. Their annual profit is described by:

P(x)= -0.000032x^2+6x-1000

Where x is measured in thousands of cases, and P(x) is measured in thousands dollars. Use differentials to estimate the change in 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

First find the derivative of P with respect to x. You can do this with the power rule.

dP/dx = -0.000032(2)x^(2-1) + (1)6x^(1-0) - (0)1000x^(0-1)

dP/dx = -0.000064x + 6

They say the sales forecast is decreased by 500 cases, do dx = -500/1000 = -0.5.

dP/(-0.5) = -0.000064(x) + 6

dP = (-0.5)(-0.000064x + 6)

The original x value was 200,000 cases, so x = 200. We can plug that into the equation.

dP = -0.5(-0.000064(200) + 6)

dP = -2.9936

P(x) is measured in thousands of dollars, so change in dollars will be dp(1000)

-$2993.6 (round to nearest dollar)

The profit will decrease by $2994