Mathematico | Mathematico

You require to build a pen out of chain-link fencing. The fencing costs $£ 40$ per meter and you have a budget of $£ 2000$ . The pen is to be rectangular in shape, with two partitions parallel to two sides of the rectangle, thus forming three separate areas.

What is the maximum possible total area for the pen?

Give your answer as an exact decimal in $m^{2}$ .

Hint

Let the height be $a$ and the width be $b$ :

Hint

In forming the pen, you use $4 a + 2 b$ meters of fencing. How many meters of fencing can you buy?

Solution

Let the height be $a$ and the width be $b$ :

So we use $4 a + 2 b$ meters of fencing.

With $£ 2000$ we can purchase $2000 / 40 = 50$ meters of fencing.

So $4 a + 2 b = 50$ , or $2 a + b = 25$ .

The area of the pen is

\begin{aligned} a b & = a (25 - 2 a) \\ = 25 a - 2 a^{2} \\ = - 2 [a^{2} - \frac{25 a}{2}] \\ = - 2 [{(a - \frac{25}{4})}^{2} - \frac{625}{16}] \\ = - 2 {(a - \frac{25}{4})}^{2} + \frac{625}{8} \end{aligned}

So the maximum area is $\frac{625}{8} = 78.125$ .