Question

A rancher needs to enclose two adjacent rectangular? corrals, one for cattle and one for sheep. if the river forms one side of the corrals and 420420 yd of fencing is? available, find the largest total area that can be enclosed.

Answer 1

Remark: the fence is clearly 420 yards, not 420420.

check the picture

The 2 adjacent rectangles (corrals) form one larger rectangle.

as shown in the figure, let x be the length of the fences perpendicular to the river, then the length of the side opposite to the river will be 420- 2x (yard)

so we can write the following function, which calculates the area A enclosed by the fence, as a function of x.

A(x)=2x(420-2x)

clearly A is a quadratic function, its graph is a parabola. This parabola looks downwards because of the minus of the term -2x

from A(x)=2x(420-2x) we can find that the x intercepts of the parabola are:

one of the roots is x=0

and

420-2x=0,  
2x=420
x=210 is the other root.

The x coordinate of the vertex is the midpoint of (0, 210), that is 210/2=105

f(105)=2*105(420-2*105)=210*(420-210=2110*210=44,100 square yard.

the highest point of the parabola is the largest value the function takes, so the maximal area of the fence.


Answer: max area= 44,100 square yard.