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?

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

----> equation A

Remember that

we have a square based box

so

----> equation B

substitute equation B in equation A

----> equation C

The volume of the box is equal to

----> equation D

substitute  equation B and equation C in equation D

solve for x

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

equate to zero

solve for x

Factor -6x

The solutions are

x=0, x=36 in

Find the value of y

so

Find the value of z

therefore

The dimensions are 36 in by 36 in by 36 in

The volume is equal to

----> is a cube

d

step-by-step explanation:

d