Optimization Help Calc 1

Question

Gloria would like to construct a box with volume of exactly 30ft^3 using only metal and wood. The metal costs $12/ft^2 and the wood costs $9/ft^2. If the wood is to go on the sides, the metal is to go on the top and bottom, and if the length of the base is to be 3 times the width of the base, find the dimensions of the box (Length, Width, Height) that will minimize the cost of construction. Round your answer to the nearest two decimal places.

Answer 1

Hi Grace,

this question has a lot of work please follow me step by step.

1- L = length W = width H = height and base,top and bottom, includes L and W and side includes W and H

2- volume = 30 = L*W*H and L = 3W given information

3- cost = 4 sides *9 + 2 base*12 = 4 W*H*9 + 2 L*W *12 = 36 H*W + 24 L*W

4- cost = 36HW + 24(3W) W = 36HW + 72W² SEE!! you have one equation with two variables,TOO BAD,now we use of volume in order to replace H

5- 30 = L*W*H = 3W*W*H = 3W²H ------> H = 30 / 3W² H = 10/ W²

6- cost = 36 (10/w² )W + 72 W² = 360/W + 72 W² Now you have equation of cost includes one variable such as W.

7- Now, it is easy, by finding derivative of cost then = zero in order to find W then much easier you find L = 3W and H = 10/w²

8- c' = -360/w² + 144w = 0 -------> w³= 5/2 -----> cube root of both side- - w = 1.36 feet

L = 3 * 1.36= 4.08 feet

H = 10/1.36²= 5.41 feet

I hope it is useful and let me know if you have any question,

Minoo