Problema Solution
My friend has $260 to spend on a fence for her rectangular garden. She wants to use cedar fencing which costs $9/meter on one side, and cheaper metal fencing which costs $4/meter for the other three sides.
What are the dimensions of the garden with the largest area she can enclose?
length for cedar side=
width for other side=
What is the largest area that can be enclosed?
area=
Answer provided by our tutors
Let
x be the length of the rectangle
y be the width of the rectangle
9*x + 4*(x + 2*y) = 260
13*x + 8*y = 260
y = (260 - 13*x)/8
We want to find the largest area that she can inclose. Since the area of the rectangle equals legth*width in our case we can write the follwoing
x*y = x* (260 - 13*x)/8 = (260x - 13x^2)/8
We need to find the maximum of the quadratic function f(x) = -(13/8)*x^2 + (260/8)*x
-(13/260) is negative the parabola opens downward and the vertex is the highest point the graph can get
The vertex of the function f(x) = -(13/8)*x^2 + (260/8)*x is x = 10 and f(10) = 325/2 = 162.5 maximum of the function. Thus we have.
length of the ceder side = 10 m
width of the other side = 16.25 m
the largest area that can be enclosed is 162.5 m^2
Note: We can also find the maximum of the function by using the first derivative of f over x.