Question

A storage box with a square base must have a volume of 90 cubic centimeters. the top and bottom cost $0.60 per square centimeter and the sides cost $0.30 per square centimeter. find the dimensions that will minimize cost. (let x represent the length of the sides of the square base and let y represent the height.

Answer 1

side length of 3.56 cm and height of 7.10 cm

Explanation:

Answer 2

For side length of 3.56 cm and height of 7.10 cm the cost will be minimum.

Solution-

Let us assume that,

x represents the length of the sides of the square base,

y represent the height.

Given the volume of the box is 90 cm³, so

`\Rightarrow V=90\\\\\Rightarrow x^2* y=90\\\\\Rightarrow y=(90)/(x^2)`

As the top and bottom cost $0.60 per cm² and the sides cost $0.30 per cm². Total cost C will be,

`C=\text{cost for top and bottom}+\text{cost for rest 4 sides}\\\\=(2x^2* 0.6)+(4xy* 0.3)\\\\=(2x^2* 0.6)+(4x* (90)/(x^2)* 0.3)\\\\=1.2x^2+ (108)/(x)`

Then,

`C'=(d)/(dx)(1.2x^2+ (108)/(x))=2.4x-\frac{108}\\\\C''=(d^2)/(dx^2)(1.2x^2+ (108)/(x))=2.4+(216)/(x^3)`

As C'' has all positive terms so, for every positive value of x (as length can't be negative), C'' is positive.

So, for minima C' = 0

`\Rightarrow 2.4x-(108)/(x^2)=0\\\\\Rightarrow 2.4x=(108)/(x^2)\\\\\Rightarrow x^3=(108)/(2.4)=45\\\\\Rightarrow x=3.56`

Then,

`y=(90)/(x^2)`

`y=(90)/(3.56^2)`

`y=7.10`

Therefore, for side length of 3.56 cm and height of 7.10 cm the cost will be minimum.