Question

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

Answer 1

Answer:

Length = Width = Height =36 inches

Volume =46,656 cubic inches

Step-by-step explanation:

Let

x ----> the length of the box-shaped in inches

y ----> the width of the box-shaped  in inches

z ---> the height of the box shaped  in inches

we know that

$x+y+z=108$

$z=108-x-y$ ----> equation A

Remember that

we have a square based box

so

$x=y$ ----> equation B

substitute equation B in equation A

$z=108-x-x$

$z=108-2x$ ----> equation C

The volume of the box is equal to

$V=xyz$ ----> equation D

substitute  equation B and equation C in equation D

$V=x(x)(108-2x)$

solve for x

$V=-2x^3+108x^2$

Since we're looking for a maximum, that will happen when the slope of the above equation is 0. And the first derivative will give us that slope.

so

calculate the first derivative

$V'=-6x^2+216x$

equate to zero

$-6x^2+216x=0$

solve for x

Factor -6x

$-6x(x-36)=0$

The solutions are

x=0, x=36 in

Find the value of y

$y=x$

so

$y=36\ in$

Find the value of z

$z=108-2(36)$

$z=108-72=36\ in$

therefore

The dimensions are 36 in by 36 in by 36 in

The volume is equal to

$V=(36)(36)(36)=46,656\ in^3$ ----> is a cube