Question

Postal regulations specify that a parcel sent by priority mail may have a combined length and girth of no more than 144 in. Find the dimensions of a rectangular package that has a square cross section and largest volume that may be sent by priority mail.What is the volume in such a package?

Answer 1

Answer:

The volume in such a package is 27648 in³

Step-by-step explanation:

Consider the provided information.

Postal regulations specify that a parcel sent by priority mail may have a combined length and girth of no more than 144 in.

Let the dimension are x by x by y.

Where x is the variable for the square base package and y is the variable for length.

Thus l=x, b=x and h=y

Then the volume of the box is: $V(x)=x^2y$ (∵V=lbh)

The maximum combined length and girth is 144.

Therefore, $4x+y=144$

$y=144-4x$

Substitute the value of y in volume of the box.

$V(x)=x^2(144-4x)$

$V(x)=144x^2-4x^3$

$V'(x)=288x-12x^2$

Substitute V'(x)=0.

$0=288x-12x^2$

$12x(24-x)=0$

$x=0\ or\ x=24$

Now apply second derivative test.

$V''(x)=288-24x$

$V''(0)=288-24(0)>0$ (Min)

$V''(24)=288-24(24)$ (Max)

Hence, the maximum volume is at x=24

If x=24 then $y=144-4(24)=48$

Substituting value of x = 24 and y = 48 $V(x)=x^2y$ gives 27648.

Hence, the volume in such a package is 27648 in³