16. Peter is building a fenced in area for his dog. He has 250 feet (ft.) of fencing to create an area for his dog. He is building it next to his house so he will only need fencing for 3 sides of the rectangular area. The diagram shows the width, d, of the area Peter is building for his dog.%0D%0A%0D%0A%0D%0A%0D%0AThe function A(d)=d(250−2d) represents the area that will be fenced in for Peter's dog, in square feet.%0D%0A%0D%0AComplete the statements.%0D%0A%0D%0A(2 points)%0D%0AA reasonable scale for the horizontal axis of the function A(d)=d(250−2d) is %0D%0A, A reasonable scale for the vertical axis of the function A(d)=d(250−2d) is

Answer 1

To determine a suitable scale for the horizontal and vertical axes of the function \( A(d) = d(250 - 2d) \), we start by analyzing the parameters of the function.

Horizontal Axis (d)

1. Domain of \( d \): Since \( d \) represents the width of the area, it must be a non-negative value. Additionally, for the fencing to be valid, the maximum value of \( d \) should be \( 125 \) ft (since \( 250 - 2d \) must also be non-negative, meaning \( 250 \geq 2d \), or \( d \leq 125 \)).
2. Range of \( d \): Thus, the reasonable values for \( d \) would be between \( 0 \) and \( 125 \).

A reasonable scale for the horizontal axis could be:

• Horizontal Axis Scale: "0 to 125 feet" with increments of 25 feet (i.e., marked at 0, 25, 50, 75, 100, 125).

Vertical Axis (A)

1. Finding the maximum area: We can find the maximum area by determining the vertex of the quadratic function. The function can be expressed in standard form \( A(d) = -2d^2 + 250d \). The vertex occurs at \( d = -\frac{b}{2a} = \frac{250}{2 \times 2} = 62.5 \) feet.
2. Calculating the Maximum Area: Substituting \( d = 62.5 \) into the area function gives \( A(62.5) = 62.5 (250 - 2 \times 62.5) = 62.5 \times 125 = 7812.5 \) square feet.

Thus, the reasonable range for the vertical axis should include the maximum area:

• Vertical Axis Scale: "0 to 8000 square feet" with increments of 1000 square feet (i.e., marked at 0, 1000, 2000, 3000, ..., 8000).

Complete Statements

• A reasonable scale for the horizontal axis of the function \( A(d) = d(250 - 2d) \) is 0 to 125 feet.
• A reasonable scale for the vertical axis of the function \( A(d) = d(250 - 2d) \) is 0 to 8000 square feet.