Write a function to represent the area as the function of the width...?

Question

Situation: Owners of a farm want to fence in an area for goats. They want to use up 40 meters of fencing they purchase.

1) What id the maximum area they can provide for the animals with 40 meters of fence if they want the pen to be a quadrilateral?

2) How does changing the width affect the area?

3) Write a function to represent the area as the function of the width?



Bonus: Now imagine that the fenced-in area is adjacent to a building. You only need to fence three sides.

Solution would be highly appreciated.

Answer 1

Wrtie the function for area as a function of w A(w).

A(w) = w * l, where 2w + 2l = 40 (they have to add up to 40), so l = 20 - w

A(w) = w* (20 - w)

Thus, A(w) = 20w - w^2

Case 1) Find the derivitive and set = 0
A'(w) = 20 - 2w = 0, thus w = 10m, and therefore l = 10m. It is a square with an area A = 100 m^2

2) Increasing or decreasing the width from 10 decreases the area exponentially.

3) A(w) = 20w - w^2

Write the function.
A(w) = w * l, with 2w + l = 40 so l = 40 - 2w
Thus A(w) = w * (40 - 2w)
The function is:

A(w) = 40w - 2w^2

Find the derivative and set = 0

A'(w) = 40 - 4w = 0 thus w = 10 and l = 20

Thus it is a rectangle 10m x 20m with an area of 200 m^2