Question

A plan for a park has a rectangular plot of wild flowers that needs to be enclosed by 54 feet of fencing. Only three sides need to be enclosed because one side is bordered by the parking lot. use Desmos to get your answers. 1. What is the largest area possible for the garden? DO NOT ROUND YOUR ANSWER. ____ squared feet2. What width will produce the maximum area? ____ feet3. What is the length of the garden that will produce the maximum area?

Answer 1

SOLUTION:

Step 1:

In this question, we are given the following:

Step 2:

a) What is the largest area possible for the garden?

Now, let the length of the rectangular plot be 54 -2x,

and the width of the rectangular plot be x,

so that:

$\begin{gathered} \text{Area = (54 -2x) x = 54 x -2x}^2 \\ \frac{dA}{dx}=\text{ 54 - 4x = 0} \\ We\text{ have that:} \\ 54\text{ = 4x } \\ \text{Divide both sides by 4, we have that:} \\ \text{x = }\frac{54}{4} \\ \text{x = 13. 5} \end{gathered}$

Then, the largest area possible for the garden will be:

$\text{Area = 54x -2x}^2=54(13.5)-2(13.5)^2=729-364.5=364.5ft^2$

b) What width will produce the maximum area?

$Width,\text{ x = 13. 5 fe}et$

c) The length of the garden that will produce the maximum area:

$\text{Length = 54 - 2x = 54 - 2( 13. 5) = 54 -27 = 27 fe}et$