Concept explainers
Enclosing a Rectangular Field David has 400 yards of fencing and wishes to enclose a rectangular area.
(a) Express the area of the rectangle as a function of the width of the rectangle.
(b) For what value of is the area largest?
(c) What is the maximum area?
To calculate:
Express the area of the rectangle as a function of width .
For what value of is the area largest.
What is the maximum area?
Answer to Problem 7AYU
Solution:
100 yards
10,000 square yards
Explanation of Solution
Given:
David has 400 yards of fencing and wishes to enclose a rectangular area.
Formula used:
Perimeter of a rectangle is twice the sum of its length and width.
Area of a rectangle is , where is the length and is the width.
Calculation:
Let the length of the rectangle be and width be .
The 400 yards of fencing can be considered as the perimeter of the rectangle.
Therefore, we get
Now, we have to find the area of the rectangle.
Thus, we get the area as
a. The area of the rectangular fence is .
b. The equation of area is a quadratic equation with . Since is negative, the vertex is the maximum point of the equation.
Thus, we get the maximum point at
Therefore, the area is maximum when the width is 100 yards.
c. The maximum area of the rectangle is
The maximum area of the rectangle is 10,000 square yards.
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