Question

close sidebar webwork/mat145f19stern/hw13/5 HW13: Problem 5 Previous Problem Problem List Next Problem (1 point) A parcel delivery service will deliver a package only if the length plus the girth (distance around, taken perpendicular to the length) does not exceed 104 inches. Find the maximum volume of a rectangular box with square ends that satisfies the delivery company's requirements.

Answer 1

Answer:

The volume in such a package is 10,415.41 in³

Step-by-step explanation:

Consider the provided information.

A parcel delivery service will deliver a package only if the length plus the girth (distance around, taken perpendicular to the length) does not exceed 104 inches.

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 104.

Therefore, $4x+y=104$

$y=104-4x$

Substitute the value of y in volume of the box.

$V(x)=x^2(104-4x)$

$V(x)=104x^2-4x^3$

$V'(x)=208x-12x^2$

Substitute V'(x)=0.

$0=208x-12x^2$

$-4x(3x-52)=0$

$x=0\ or\ x=\frac{52}{3}$

Now apply second derivative test.

$V''(x)=208-24x$

$V''(0)=208-24(0)>0$ (Min)

$V''(\frac{52}{3})=208-24(\frac{52}{3})$ (Max)

If x=52/3 then $y=104-4(\frac{52}{3})=\frac{104}{3}$

Substitute x = 52/3 and y = 104/3 in $V(x)=x^2y$

$V(x)=(\frac{52}{3})^2\times \frac{104}{3}=10,415.41$

Hence, the volume in such a package is 10,415.41 in³