Question

10 a farmer is going to fence in an area next to the barn into a rectangle split into two equal pens. the farmer has 165 yards of fencing available.
(a) define the function ( a(w) ) to represent the fenced in area as a function of the width. explain your reasoning.
(b) determine the maximum area of the entire fenced in area as well as the width and length that will result in the maximum area. explain your reasoning.

Explanation:

for (a):

Step1: Define variables

Let width = \( w \), length = \( l \). Fencing: \( 3w + l = 165 \).

Step2: Express length in terms of width

\( l = 165 - 3w \).

Step3: Area function

Area \( A(w) = w \times l = w(165 - 3w) = -3w^2 + 165w \).

for (b):

Step1: Identify quadratic form

\( A(w) = -3w^2 + 165w \) (a < 0, opens downward).

Step2: Find vertex width

Vertex at \( w = -\frac{b}{2a} = -\frac{165}{2(-3)} = 27.5 \).

Step3: Calculate length

\( l = 165 - 3(27.5) = 82.5 \).

Step4: Compute max area

\( A(27.5) = 27.5 \times 82.5 = 2268.75 \).

Answer:

a) \( A(w) = -3w^2 + 165w \)
b) Maximum area: 2268.75 square yards; width: 27.5 yards; length: 82.5 yards