Chapter 3.4, Problem 15AYU

Constructing Rain Gutters A rain gutter is to be made of aluminum sheets that are 12 inches wide by turning up the edges ${90}^{\circ }$. See the illustration.

(a) What depth will provide maximum cross-sectional area and hence allow the most water to flow?

(b) What depths will allow at least 16 square inches of water to flow?

To determine

To calculate:

What depth will provide maximum cross sectional area and hence allow the most water to flow.

What depth will allow at least 16 square inches of water to flow?

Answer to Problem 15AYU

Solution:

3 inch.

Between 2 inch and 4 inch.

Explanation of Solution

Given:

A rain gutter is to be made of aluminium sheets that are 12 inches wide by turning up the edges 90 degrees.

Formula used:

The area of a rectangle is the product of its length and breadth.

Calculation:

From the given picture, we can see that the length of the rectangle is $l=12-2x$ and the breadth is x.

Therefore, the area is A=lb=(12-2x)x=12x-2x^2.

The area of the rectangle is a quadratic equation with $a=-2,b=12,c=0$.

Here, since a is negative, the vertex is the point of maximum.

Therefore, in order to have the maximum cross sectional area, the depth of the sheet should be maximum.

Thus, the maximum depth of the sheet is

$\frac{-b}{2a}=\frac{-12}{2(-2)}=3$

(a) Therefore, in order to have the maximum cross sectional area, the depth should be 3 inches.

(b) when $A=16$, we get

$A\Rightarrow 12x-2x^2=16$

$\Rightarrow x^2-6x+8=0$

$\Rightarrow (x-4)(x-2)=0$

$\Rightarrow x=2$

and

$\Rightarrow x=4$

Thus, for at least 16 square inches of water to flow, the depth should be between 2 inches and 4 inches.