Quadratics?

Question

Seth has a piece of aluminum that is 10 in. wide and 12 ft. long. He plans to form a gutter with a rectangular cross section and an open top by folding up the sides. What dimensions of the gutter would maximize the amount of water that it can hold? Hint: Write a quadratic function for the cross-sectional area and find the vertex.

Also, I am not in Calculus, so could someone please tell me how to solve without using any "Calculus terms"? Thanks!

Answer 1

I'll show you a non-calculus way that uses quadratics and parabolas.

First, we don't need to know how long the gutter is, it doesn't matter unless you need to calculate the total volume it will hold. For now we just need to find the cross-sectional area that gives this maximum.

Let's call the height of the gutter h and the width w. the total dimensions of the gutter will be

2h+w=10: this is the constraint equation. If we take a thin slice of the gutter to find it's volume it's really like finding the area, so we want to maximize area A=hw.

so now using the two equations we have A=h(10-2h)
=10h-2h^2. Let's complete the square

A=-2(h^2-5h+25/4)+25/2=-2(h-5/2)^2+25/2

in canonical form
A-25/2=-2(h-5/2)^2

You can see that when h=5/2=2.5 you'll get the largest area.

At this value of h=5/2, w=10-2*(5/2)=5
the area is 5*5/2=25/2, which if you look again at the canonical form you see this value.

Let's check by taking another larger and smaller value of h and see if the area goes up or down.

Test 1: h=3, in this case w=4 and the area is 12<12.5. So test 1 is good.

Test 2 h=2, in this case w=6 and again the area is 12<12.5. So we know the dimensions that will give the maximum volume are h=2.5 and w=5

Answer 2

We've got 10in X 144 in.
Let the height of the proposed gutter be 'x' inches.
Then, sides of the cross-sectional rectangle are:
(10-2x)X(144-2x)

Volume of gutter: (10-2x)*(144-2x)*x
V =4x(x-5)(x-72)
=4(x^3-77x^2+360x)
The only method I can think of is calculus, so here goes:

dV/dx=4(3x^2-154x+360)
For maxima: dv/dx=0
x=48.4 or x=2.45

As x can't be as high as 48.4, x=2.45in

Dimensions of gutter: 2.45in X 5.1in X 139.1in