Consider a closed rectangular box with a square base with side x and height y .

Question

a. Find an equation for the surface area of the rectangular box.

S(x,y)=______________

b. If the surface area of the rectangular box is 32 square feet, find dy/dx when x=2 feet and y=3 feet.

dy/dx=______________

Answer 1

S(x,y) = 4xy + 2x^2

32 = 4xy + 2x^2

4xy = 32-2x^2

y = 32/4x - (1/2)x

y' = dy/dx = -8/x^2 - 1/2 = -8/4 - 1/2 = -2 - 1/2 = -5/2

y'= -5/2

or

32 = 4xy + 2x^2

0 = 4xy' + 4y + 4x = 4(2)y' + 4(3) + 4(2) = 8y' +20

8y'=-20

y'= -20/8 = -5/2

dy/dx = -5/2