In these problems you are asked to find a function that models a real$-$life situation and then use the model to answer questions about the situation. Use the Guidelines for Modeling with Functions to help you answer these questions.

Consider the following problem: A farmer has $3,600$ feet of fencing and wants to fence off a rectangular field that borders a straight river. The farmer does not need a fence along the river $($see the figure$).$

$$

What are the dimensions of the field of largest area that can be fenced? $($Let $x$ be the width of the field in feet and $\frac{?}{?}$ be the length of the field in feet.$)$

$($a$)$ Experiment with the problem by drawing several diagrams illustrating the situation. Calculate the area of each configuration, and use your results to estimate the dimensions of the largest possible field $($in feet$).($Round your answers to the nearest hundred feet.$)$

$x=,ft$

$1=,ft$

$($b$)$ Find a function that models the area of the field in terms of one of its sides.

$A(x)=$

$$

$($c$)$ Use your model to solve the problem, and compare with your answer to part $($a$).($Enter your answers in feet$).$

$x=,ft$

$1=,ft$