Question

An enclosed area is designed using one existing wall, and 72m of fence material. What is the maximum area of the enclosure?

a.600 squared meters.
b.640 squared meters.
c.648 square metres.
d.656 square metres ​

Answer 1

Answer:

The maximum area possible is 648 squared meters.

Step-by-step explanation:

• Let the length of the existing wall be $\ell$
• And let the width of the fence be w.

●The area of the enclosure will be given by:

$A=w\ell$

Since the area is bounded by one existing wall, the perimeter (the 72 meters of fencing material) will only be:

• $72=2w+\ell$

We want to maximize the area.

From the perimeter, we can subtract 2w from both sides to obtain:

• $\ell=72-2w$

Substituting this for our area formula, we acquire:

• A=w(72-2w)

This is now a quadratic. Recall that the maximum value of a quadratic always occurs at its vertex.

We can distribute:

• $A=-2w^2+72w$

Find the vertex of the quadratic. Using the vertex formula, we acquire that:

• $\displaystyle w=-\dfrac{b}{2a}=-\dfrac{(72)}{2(-2)}=18$

So, the maximum area is:

• $A=-2(18)^2+72(18)=648\text{ meters}^2$