SOLUTION: A 24-inch-wide sheet of metal is to be bent into a rectangular trough with the cross section shown in the illustration. Find the dimensions that will maximize the amount of water t

Question 565891: A 24-inch-wide sheet of metal is to be bent into a rectangular trough with the cross section shown in the illustration. Find the dimensions that will maximize the amount of water the trough can hold. That is, find the dimensions that will maximize the cross-sectional area.(depth and width = 24 inches)
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!

Before I even start, I want you to go back and read the question as you posted it. "...shown in the illustration." Right. This is Algebra.com -- NOT the Psychic Hot Line. First lesson: Mathematics requires using your head for something besides a hat rack.

Be that as it may, I'm going to presume that you want to fold up two sides making an unknown depth (equal on both sides) on a trough with no top and a bottom that perforce measures 24 inches minus 2 times the depth. To the extent that my assumptions about the true nature of your problem are correct, begin with the concept that the area of a rectangle is given by length (or depth in this case) times width.

For this problem we let

represent the amount folded up or the depth of the trough, and then the width of the rectangle (the bottom of the trough must be

. Multiplying length times width gives area as a function of the depth:

Written in standard

form:

Algebra Solution Recognize that this is a quadratic polynomial function that will graph to a parabola. Since the lead coefficient is negative, the parabola must open downward, hence the vertex is a maximum. Using