Complete parts a through f below to find nonnegative numbers x and y that satisfy the given requirements. Give the optimum value of P. x plus y equals 81 and Pequalsx squared y is maximized a. Solve x plus y equals 81 for y. yequals 81 minus x b. Substitute the result from part a into the equation Pequalsx squared y for the variable that is to be maximized. Pequals x squared left parenthesis 81 minus x right parenthesis c. Find the domain of the function P found in part b. left bracket 0 comma 81 right bracket (Simplify your answer. Type your answer in interval notation.) d. Find StartFraction dP Over dx EndFraction. Solve the equation StartFraction dP Over dx EndFraction equals0. StartFraction dP Over dx EndFraction equals nothing

y = 81-x the domain of P (x) is [0, 81] P is maximized at (x, y) = (54, 27)

Step-by-step explanation:

Given

x plus y equals 81 x and y are non-negative

Find

P equals x squared y is maximized

Solution

a. Solve x plus y equals 81 for y.

y equals 81 minus x

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b. Substitute the result from part a into the equation P equals x squared y for the variable that is to be maximized.

P equals x squared left parenthesis 81 minus x right parenthesis

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c. Find the domain of the function P found in part b.

left bracket 0 comma 81 right bracket

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d. Find dP/dx. Solve the equation dP/dx = 0.

P = 81x² - x³

dP/dx = 162x - 3x² = 3x (54 - x) = 0

The zero product rule tells us the solutions to this equation are x=0 and x=54, the values of x that make the factors be zero. x=0 is an extraneous solution for this problem so ...

P is maximized at (x, y) = (54, 27).