Question

The riding stables just received an unexpected rush of registrations for the next horse show, and quickly needs to create some additional paddock space. There is sufficient funding to rent 1200 feet of temporary chain-link fencing. The plan is to form two paddocks with one shared fence running down the middle. What is the maximum area that the stables can obtain, and what are the dimensions of each of the two paddocks?a. 100b. 150c. 200d. 220e. 250

Answer 1

Answer:

Maximum added area will be 60,000 ft²

Each paddock measure 200 ft by 150 ft, and the paddocks share a 200-ft long side.

Step-by-step explanation:

Consider the provided information.

There is sufficient funding to rent 1200 feet of temporary chain-link fencing.

The plan is to form two paddocks with one shared fence running down the middle.

The total length of fencing is 1200 feet that is same as the total perimeter including that shared line down the middle.

Therefore, the perimeter is

$P = 1200 = 2L + 3w$

Solve for L

$1200-3w = 2L$

$L=600-\frac{3w}{2}$

As we know the area of rectangle is A=L×W

Substitute the value of L in above formula.

$A=(600-\frac{3w}{2})\times w\\A=600w-\frac{3w^2}{2}$

$A=-\frac{3w^2}{2}+600w$

The above equation makes a downward parabola.

In order to find the maximum, find the vertex of parabola as shown:

$W=\frac{-b}{2a}=\frac{600}{2\times\frac{3}{2}}=200$

Hence, the width should be 200 ft in order to get maximum area,

Therefore, the length will be:

$1200 = 2L + 3(200)$

$L=300$

Now substitute the value of L and W in area formula.

A=300×200=60,000

Hence, maximum added area will be 60,000 ft²

Hence, the length should be 300 ft, and each paddock should then be 150 feet long.

Each paddock measure 200 ft by 150 ft, and the paddocks share a 200-ft long side.