Question

ppose an airline policy states that all baggage must be box-shaped with a sum of length, width, and height not exceeding 108 in. What are the dimensions and volume of a square based box with the greatest volume under theses conditions

Answer 1

dimensions are 36 x 36 x 36

Volume= 46656 inch³

Explanation:

Supposing the dimensions x , x and y as the base is a square base.

So, their sum is : x + x + y= 108

2x + y= 108

y= 108 - 2x --> eq(1)

Also, they are lengths, x,y >0

Next is to determine the maximum volume, the function is

V(x,y)= x.x.y => x²y

By subsitituing value of y from eq(1) in above equation

V(x)=x²(108-2x)

V(x)= 108x² - 2x³

In order to find critical point, take the derivative and set it to zero

V'(x)= 216 x - 6x² = 0

36 - x =0

x = 36 in

Plugging the value of 'x' in eq(1)

(1)=> y = 108 - 2(36)

y= 36 in

Therefore, the volume of a square based box will be

V= x²y = 36² . 36

V=46656 in³