Related rates calculus

Question

A small closed rectangular box is to have a capacity of 72 cm^3. If the box is to be twice as long as it its wide, determine the dimensions of the box that uses the least material.

Answer 1

You will need to use the volume and surface area equations for a rectangle.

V=L*W*H

SA=2(L*W)+2(W*H)+2(L*H)

you will need to draw a diagram of the situation. draw a box label a width as X and a length of 2X and that the entire volume is 72 cm^3

Plug in the 72 cm^3, L, & W into the volume equation and solve for H

H= 36/x^2

Plug that into the surface area and reduce down.

SA=4X^2 +216/X

now take the derivative of the equation and set equal to 0.

solve for x and find the width and length and height of the box.

W=3 L=6 H=4