Question

A farmer needs to enclose three sides of a field with a fence (the fourth side is a river). The farmer has 49 yards of fence and wants the field to have an area of 294 sq-yards. What should the dimensions of the field be? (For the purpose of this problem, the width will be the smaller dimension (needing two sides); the length with be the longer dimension (needing one side). Additionally, the length should be as long as possible.)

Answer 1

Answer:

The length of the field = 24.5 yards

The width of the field = 12 yards.

Step-by-step explanation:

If "w" is the width, then the length is 49 - 2w.

The area of the rectangle field = length × width

= w(49 - 2w)

Area = $49w - 2w^2$

This a quadratic equation, the vertex of x coordinate is w

w = $\frac{-b}{2a}$

Here a = -2 and b = 49

w = $\frac{-49}{2(-2)} = \frac{-49}{-4} = 12.25$.

So width of the field is 12.25 yards.

The length of the filed = 49 - 2(12.25) = 24.5

Area = 12.25 × 24.5 = 300.125 square yards.

The field has an area of 294.

Therefore, the length must be 24.5 yards and width must be 12 yards.

24.5 × 12 =294 square yards.

Therefore, the length of the field = 24.5 yards

the width of the field = 12 yards.