Math $251$ SLO$3$

Student Name:

Instructor Name:

This assignment will lead you through an optimization problem. The problem we are going to be working through is as follows:

A rain gutter is to be constructed from a metal sheet of width $45$ cm by bending up onthird of the sheet on each side through an angle of $$ radians. How should $$ be chosen so that the gutter will carry the maximum amount of water?

To assist with this problem, we provide the following sketch of the situation. Label this graph with the side lengths $(15cm)$ and the angles $().$

Our goal is to find a formula for the area of the shaded region in the sketch above. This region can be decomposed into three pieces. One rectangle in the middle and two identical triangles on the sides. Express the area of each piece in terms of the angle $($in radians$).$ Note: standard formulas for the area of a triangle $A=\frac{1}{2}bh$ and rectangle $A=bh.$

$A_{\text{t}\text{r}\text{i}}=$

$A_{\text{r}\text{e}\text{c}\text{t}}=$

We then add these three areas together to create the area function we are trying to maximize.

$A()=$

What restrictions will be on the angle $($in radians$)?$

Differentiate the function $A()$ to obtain the derivative.

$A^{\text{'}}()=$

Use your calculator to approximate any zeros of the first derivative within the restrictions placed on $$ in problem $4.$

$=$

Create a number line and apply the first derivative test to determine the angle $$ we should choose to obtain the maximum area.

State the conclusion of the problem. I.E$.$ answer the question, "How should $$ be chosen so that the gutter will carry the maximum amount of water?"