SOLUTION: Hi, I am working on a project regarding applications of polynomial functions for my college algebra class. I am stuck on one particular questions and the tutor I spoke to on ca

Click here to see ALL problems on Polynomials-and-rational-expressions

Question 243211: Hi,
I am working on a project regarding applications of polynomial functions for my college algebra class. I am stuck on one particular questions and the tutor I spoke to on campus approached this question in a manner that doesn't make sense to me.
I will start by pasting the questions below:
The USPS requires that any package submitted for shipment satisfy the condition that the girth plus the length must be less than or equal to 106 inches. The length is always the longest dimension, and the girth is the perimeter of a cross section perpendicular to the length. If the cross section is rectangular, of width w and height h, then the girth is
2w+2h.
a) Suppose the cross section perpendicular to the length is indeed a square.
Construct a polynomial function for the volume of the package, in terms of the
length of the sides of the square cross section.
b. By graphing the function or by using a table, estimate the dimensions of the
package of the largest volume that can be shipped.
If I am understanding this question correctly the actual dimensions of this box would be equal since it is a square at the point perpendicular to the length which would be twice as long as any one individual side. Since the overall dimensions can't exceed 106 inches, here is my approach. The girth is a square which to me is 4(w) + 2 (w)=length, so 106/6 =17.66666667 inches.
On part b. they ask for the maximum volume possible. Since volume = L x W x H
I came up with 17.66666667 x 17.66666667 x 2(17.66666667) which gave me a maximum voolume of 11027.92593 cubic inches.
My tutor on campus approached it as follows:
girth + length can't exceed 106 inches
girth = 2W +2L
girth= 2W+ 2L= 4W
4W + H can't exceed 106 inches
H can't exceed 106-4W
V=106 W-4W^3
Maximum 210 cubic inches when w=3
This doesn't make sense to me and it has to be addressed in a polynomial form.
Thanks for your help since I an really stumped on this one.
BK
Answer by scott8148(6628) (Show Source):
You can put this solution on YOUR website!
a) width = w ___ depth = w ___ girth = 4w
___ length = 106 - 4w
___ volume = width * depth * length = w * w * (106 - 4w) = 106w^2 - 4w^3

b) see graph