a rectangular package to be sent by a postal service can have a maximum combined length and girth (perimeter of a cross section)

Question

2 years ago

6

of 120 inches (see figure). find the dimensions of the package of maximum volume that can be sent. (assume the cross section is square.)

1 answer:

satela [25.4K] · Answer 1 · 2024-01-04T02:42:38+03:00

The dimensions of the package of maximum volume that can be sent are 20 inches by 40 inches.

Let us represent the length of the square cross-section of the postal package with x, and the width of the package with y.

So, the perimeter is given by the equation -

y + 4x = 120

y = 120-4x -----------(1)

the volume will be

V = $x^{2} y$ -----------(2)

Now, substitute (1) in (2), and we get,

V = $x^{2}$ (120-4x)

V = 120 $x^{2}$ - 4 $x^{3}$

Differentiating with respect to x, we get,

V' = 240x - 12 $x^{2}$

V'=0

hence, 240x - 12 $x^{2}$ = 0

12 $x^{2}$ = 240x

dividing by 120x on both sides,

hence, x=20

Now, we know that,

y = 120 - 4x

y = 120-4(20)

y = 120-80

y=40

Hence, the dimensions that maximize the volume of the package are 20 inches by 40 inches.