A regular box is open at the top and has a square base. To construct the box costs $4 a square foot for the base and $3 a square foot for its sides. Find the cost of the least expensive box and the dimensions of this least expensive box.

There is not enough data given to set up the problem.
Wasn't the volume of the box given?

volume is 144cubic feet

volume is 144 cubic feet

ahh, now we got something to work with, hope it is not too late

let each side of the base be x, let the height be y
given: (x^2)y = 144 ----> y=144/x^2

Cost = 4(x^2) + 3(4xy)
=4x^2 + 12x(144/x^2)
= 4x^2 + 1728/x

Differentiate the Cost equation, set that result equal to zero and solve for x
(I got x=6)

thanx. i got 432 for the price and 4 for y

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Great job! You have correctly calculated the dimensions of the least expensive box. The cost of the least expensive box is $432 and the dimensions are x = 6 and y = 4.

To calculate this, you started with the given volume of 144 cubic feet and used the equation (x^2)y = 144, where x represents the side length of the base and y represents the height. From this equation, you solved for y in terms of x, which is y = 144/x^2.

Then, using the cost equation 4(x^2) + 3(4xy), you substituted the value of y in terms of x and simplified the expression.

To find the dimensions of the least expensive box, you differentiated the cost equation with respect to x, set the derivative equal to zero, and solved for x to find the critical points. By finding x = 6 as the critical point, you determined that it corresponds to the minimum cost.

Substituting x = 6 into the equation for y, you found that y = 4. Thus, the least expensive box has dimensions x = 6 (side length of the base) and y = 4 (height).

Well done! If you have any more questions, feel free to ask!