Question

Juan is building a fenced-in area for his pet ostrich. He has 450 feet (ft.) of fencing to create an area for his ostrich. He is building it next to his house so he will only need fencing for 3 sides of this rectangular area.
(2 points)
The function A(d) = d (450 - 2d) represents the area that will be fenced in for Juan's ostrich, in square feet.
Complete the statements:
A reasonable scale for the horizontal axis of the function A(d) = d(450 - 2d) is ___
A reasonable scale for the vertical axis of the function A(d) = d(450 - 2d) is ___

Answer 1

To determine a reasonable scale for the horizontal and vertical axes of the function \( A(d) = d(450 - 2d) \), we should first understand the context and the limits of the variables involved.

### Horizontal Axis (d):
- The variable \( d \) represents the width of the fenced-in area. Since Juan has a total of 450 feet of fencing and is using it for 3 sides of the rectangular area, the maximum value of \( d \) occurs when \( d \) is at its largest possible value and is constrained by \( 450 - 2d \) needing to be positive.

- Setting \( 450 - 2d > 0 \):
\[
450 - 2d > 0 \\
450 > 2d \\
225 > d
\]

Thus, \( d \) can range from \( 0 \) to \( 225 \) feet.

### Reasonable Scale for Horizontal Axis:
A reasonable scale could be from \( 0 \) to \( 225 \), perhaps in increments of \( 25 \) feet:
- **Horizontal Axis Scale**: 0 to 225 with increments of 25.

### Vertical Axis (A(d)):
- The function \( A(d) = d(450 - 2d) \) gives the area in square feet. To find the maximum area, we can analyze the area function further.

- The area is a quadratic function, which opens downwards, \( A(d) = -2d^2 + 450d \). The vertex of this parabola, where the maximum area is achieved, is found using the vertex formula \( d = -\frac{b}{2a} \) for the standard form \( A(d) = ad^2 + bd + c \):
\[
d = -\frac{450}{2(-2)} = \frac{450}{4} = 112.5
\]

- Now we find \( A(112.5) \):
\[
A(112.5) = 112.5(450 - 2(112.5)) = 112.5(450 - 225) = 112.5(225) = 25312.5 \text{ square feet}
\]

### Reasonable Scale for Vertical Axis:
Since the maximum area \( A \) occurs at \( 25312.5 \) square feet, a reasonable vertical scale could be from \( 0 \) to \( 26000 \) square feet, in increments of \( 5000 \) square feet:
- **Vertical Axis Scale**: 0 to 26000 with increments of 5000.

### Summary:
- **Horizontal Axis Scale**: 0 to 225 with increments of 25.
- **Vertical Axis Scale**: 0 to 26000 with increments of 5000.