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

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

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

To find the value of x that minimizes the cost, we need to find the minimum of the cost equation. Let's go step by step to simplify the equation and find its derivative correctly.

First, let's write the cost equation:

Cost = (1 * area of glass top) + (0.5 * area of sides and base)

The area of the glass top is x^2.

The area of the sides and base consists of the sides (4 sides with a length of x) and the base (x * x), so it is 4x^2 + x^2 = 5x^2.

The total cost equation is: Cost = (1 * x^2) + (0.5 * 5x^2) = x^2 + 2.5x^2 = 3.5x^2.

Now, let's find an equation for volume:

Volume = length * width * height

Since it is a rectangular box with a square base, the length and width are both x, and the height is given as y, which is equal to y = 21 / x^2.

So, the volume equation is: 21 = x * x * (21 / x^2) = 21x / x^2 = 21 / x.

Now, let's differentiate the cost equation with respect to x:

d(Cost)/dx = d(3.5x^2)/dx

Differentiating term by term, we get:

d(Cost)/dx = 2 * 3.5 * x^(2-1) = 7x.

Setting the derivative equal to zero, we have:

7x = 0

x = 0

Ignoring the x = 0 solution as it doesn't make sense in this context, we need to find the critical points by solving the volume equation for x.

21 / x = 0

This equation has no solution, so there are no critical points within the given constraint.

Therefore, there is no minimum value for the cost within the given constraint.