Question

A farmer decides to enclose a rectangular​ garden, using the side of a barn as one side of the rectangle. What is the maximum area that the farmer can enclose with 2020 ft of​ fence? What should the dimensions of the garden be to give this​ area?

Answer 1

the farmer is going to construct a rectangular shaped barn with dimensions

a  and  b, 

such that 2a+2b=2020, (the perimeter is 2 width+2 length)

dividing by 2, we have a+b=1010

writing b in terms of a, we have b=1010-a

so f(a)=a(1010-a) is the function which calculates the area depending on a.

for example the area enclosed is the dimensions are a= 10, b=1000

is f(10)=10*(1010-10)=10*1000=10,000 square feet.


f(a) is a polynomial function of degree 2, that is a quadratic.

The graph of a quadratic polynomial is a parabola, in our case, a parabola which opens downward since the sign of the coefficient of the largest term of the quadratic, is negative.

since the quadratic is given in factorized form, we easily find that the roots are a=0 and a=1010

since f(0)=0 and f(1010)=0.

in the middle of these points, we have 1010/2=505, which is the x-coordinate of the vertex.

f(505) gives us the highest point, which is f(505)=505(1010-505)=$505^{2}$=255,025 ft squared.

the highest point of the parabola, represents the largest value of f so the largest possible area.


Answer: 255,025 ft squared.