To determine the width that gives the maximum area for the rectangular dog pen with 52 feet of fencing, and what that maximum area is, we can use the principles of quadratic functions.
First, let’s denote the width by \( x \). The perimeter of the rectangle is 52 feet, so the length \( y \) can be expressed as:
\[ 2x + 2y = 52 \]
\[ y = 26 - x \]
The area \( A \) of the rectangle can be expressed as:
\[ A = x \cdot y \]
\[ A = x \cdot (26 - x) \]
\[ A = 26x - x^2 \]
This quadratic equation \( A = -x^2 + 26x \) is a parabola that opens downwards. The maximum area occurs at the vertex of the parabola.
For a quadratic equation in the form \( A = ax^2 + bx + c \), the x-coordinate of the vertex is given by:
\[ x = -\frac{b}{2a} \]
In this case, \( a = -1 \) and \( b = 26 \):
\[ x = -\frac{26}{2(-1)} \]
\[ x = \frac{26}{2} \]
\[ x = 13 \]
So, the width that maximizes the area is 13 feet. Now, let’s find the maximum area corresponding to this width:
\[ y = 26 - x = 26 - 13 = 13 \]
\[ A = 13 \cdot 13 = 169 \text{ square feet} \]
Thus, the width that gives the maximum area is 13 feet, and the maximum area is 169 square feet.
So, the correct answer is:
**width = 13 ft; area = 169 ft²**
17.
Suppose you have 52 feet of fencing to enclose a rectangular dog pen. The function , where x = width, gives you the area of the dog pen in square feet. What width gives you the maximum area? What is the maximum area? Round to the nearest tenth as necessary.
width = 13 ft; area = 507 ft2
width = 13 ft; area = 169 ft2
width = 26 ft; area = 169 ft2
width = 26 ft; area = 364 ft2
1 answer
1 answer