Please explain every step thanks for their product. You working on what to use a
ID: 2886631 • Letter: P
Question
Please explain every step thanks
for their product. You working on what to use and what sizes, given that the volume of the box must be 512 inches Find a formula for the surface area of the closed, rectangular box, with a square base x inches by x inches and height h inches. Simplify your answer as much as possible. (I suggest drawing a picture!) Find the dimensions giving the minimum amount of material (minimum surface area) of the closed rectangular box. You must use optimization methodology to solve this problem and you must show that this is a minimum. (Round your answer to 5 decimal places, if necessary) a. b. C.Explanation / Answer
Volume of box = 512 in3
(a). Length of square base = x
Height of box = h
Volume = x*x*h
x^2 * h = 512
h = 512 / x^2 ..............(1)
Surface area of box = 2x^2 + 4*x*h
Putting the value of h from equation (1):
Surface area = 2x^2 + 4*x*512 / x^2
S = 2x^2 + 2048 / x
(b). For surface area to be minimum,
Putting, dS / dx = 0
d/dx (2x^2 + 2048 / x) = 0
4x - 2048 / x^2 = 0
4x^3 = 2048
x^3 = 512
x = 8 in
Putting x = 8 in equation (1):
h = 512 / 8^2
h = 8 in
(c). Minimum surface area = 2*(8)^2 + 2048 / 8
= 128 + 256
= 384 in^2