Question

A display case is in the shape of a rectangular box with a square base. Suppose the volume
is 21 cubic ft and it costs $1 per square ft. to build the glass top and $0.50 per sq. ft. to
build the sides and base. If x is the length of one side of the base, what value should x have
to minimize the cost? Give your answer to two decimal places.

i made the equation for the cost and got x^2 + .5x^2 + .5xy then i plugged in what i got for y when i made a volume equation. y=21/x^2 and then i took the derivative and got 2x + x -10.5/x^2 but i don't know if that's right and where to go from there

Answer 1

if the box as height h, then the cost is

c = x^2 + .5*4xy + .5x^2
= 3/2 x^2 + 2xy
= 3/2 x^2 + 2x(21/x^2)
= 3/2 x^2 + 42/x

dc/dx = 3x - 42/x^2
= 3(x^3-14)/x^2

dc/dx=0 when
x^3 = 14
x = ∛14

Answer 2

thanks! i forgot to add the 4 to the .5xy