Graph the volume function
in Algebra 2 Answers by

Your answer

Your name to display (optional):
Privacy: Your email address will only be used for sending these notifications.
Anti-spam verification:
To avoid this verification in future, please log in or register.

1 Answer

The total surface area of a cube is 6a^2, where a is the length of its side. A cube produces the maximum volume. If the area of the paper is completely used to form a cube 6a^2=11*8.5 and a=3.9476". The maximum volume is a^3=61.52 cu in approx.

If, however, the area of the paper could be used to cover a sphere of radius r, then 4πr^2=11*8.5, and r=√11*8.5/4π=2.7277" producing a sphere with volume=4πr^3/3=85.01 cu in approx.

A more practical solution consists of folding the paper to form an open box. If we use L, W and H to represent the length, width and height of the box we can write: volume V=LWH; W+2H=8.5, L+2H=11, so we can express L and W in terms of H: W=8.5-2H, L=11-2H. V=H(11-2H)(8.5-2H)=93.5H-39H^2+4H^3.

dV/dH=93.5-78H+12H^2=0 at a maximum or minimum. The next derivative gives us -78+24H.

First, solve the quadratic: H=(78±√(78^2-4*12*93.5))/24=(39±√399)/24=4.9146" or 1.5854". By substituting these values into the second derivative, we can see that H=1.5854" is a maximum, while H=4.9146" is a minimum, and results in negative W and V. From H we can find L=11-2H=7.8292" and W=5.3292", V=66.1482 cu in. The paper has 4 squares side 1.5854" cut out from the corners. The paper then folds up to make an open box with the volume V.

On the graph the red curve is the volume function V(H). The maximum is clear to see at 66.1482 cu in. The sloping lines show the width and length (W(H) in blue and L(H) in green), both of which must be positive to form a box with H. So the volume only has meaning when V, H, W and L are all positive (all in quadrant 1). When V is at a maximum, H, W and L are all positive. H is positive by being to the right of the origin and V, W and L are positive because they're all above the H-axis.

by Top Rated User (696k points)

Related questions

1 answer
asked Mar 15, 2011 in Algebra 1 Answers by anonymous | 3.2k views
0 answers
Welcome to, where students, teachers and math enthusiasts can ask and answer any math question. Get help and answers to any math problem including algebra, trigonometry, geometry, calculus, trigonometry, fractions, solving expression, simplifying expressions and more. Get answers to math questions. Help is always 100% free!
84,130 questions
89,068 answers
6,775 users