Math Help Forum: Maximum Volume

I've spent hours trying to figure this out.
If anyone could quickly help me out, I'd be very greatful.
If possible, I need an answer by Tomorrow.
Thanks in Advance: Glenn Brown

Maximum Volume: An open box with locking tabs is to
be made from a square piece of material 24 inches on a
side. This is to be done by cutting equal squares from the
corners and folding along the dashed lines shown in the
figure.



(a) Verify that the volume of the box is given by the
function V(x) = 8x(6 - x)(12 - x).
(b) Determine the domain of function V.
(c) Sketch a graph of the function and estimate the
value of x for which V(x) is is maximum.

Originally Posted by xglennx

I've spent hours trying to figure this out.
If anyone could quickly help me out, I'd be very greatful.
If possible, I need an answer by Tomorrow.
Thanks in Advance: Glenn Brown

Maximum Volume: An open box with locking tabs is to
be made from a square piece of material 24 inches on a
side. This is to be done by cutting equal squares from the
corners and folding along the dashed lines shown in the
figure.



(a) Verify that the volume of the box is given by the
function V(x) = 8x(6 - x)(12 - x).

volume = length*width*height.

what is the length? (24 - 2x), since we take the side that is 24 inches and fold away x inches from one side and x inches from the other.

what is the width? (24 - 4x), since we take a side that is 24 inches and fold away 2x from each side.

what is the height? this is simply x, for obvious reasons.

multiply those and simplify to get the volume.

(b) Determine the domain of function V.

the domain of a function is the set of input values (in this case, x-values) for which the function is defined. this is a polynomial, what is the domain for all polynomials?

(c) Sketch a graph of the function

find the x- and y-intercepts. after finding the x-int, divide the x-axis into regions with the intercepts as the dividing line. test x-values from each region to see whether the graph is above the x-axis or below the x-axis.

also, find the end behavior. one end will go to infinity, while the other goes to -infinity. which which end does which. after this, you will be able to draw the graph (it will be in the general shape of an graph. you should know what that looks like, if you don't, look it up)

and estimate the
value of x for which V(x) is is maximum.

how accurate does your estimate have to be? are you allowed to use calculus?

yes. this course is an AP pre-cal class. that question, for some reason, has me confused. generally, the graph doesnt have to be accurate, like a hand drawn grid on a notebook paper. thanks for your help ^^^ your input was a great help! -glenn