To determine a reasonable scale for the horizontal and vertical axes of the function \( A(d) = d(250 - 2d) \), we need to analyze the possible values for \( d \) (the width) and the corresponding values for \( A(d) \) (the area).
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Horizontal Axis (d):
- The variable \( d \) represents the width of the rectangular area and must meet the condition where the total fencing used does not exceed 250 ft. Since the length uses up \( 250 - 2d \) feet of fencing and \( d \) must be positive, \( d \) must be at a minimum of 0 and at a maximum such that \( 2d < 250 \).
- Therefore, \( d \) can range from 0 to 125 ft.
- A reasonable scale for the horizontal axis could be increments of 10 ft, since it allows for clear labeling and plotting. Thus, ranges from 0 to 130 ft would be suitable for visualization.
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Vertical Axis (A(d)):
- The function \( A(d) = d(250 - 2d) \) is a quadratic equation, which opens downwards, representing a parabolic shape. To find the maximum area, we can derive the expression:
- The area \( A \) will be maximized when \( d \) is at its vertex. The vertex of a parabola in the form \( A(d) = -2d^2 + 250d \) occurs at \( d = -\frac{b}{2a} = -\frac{250}{2 \times -2} = 62.5 \) ft.
- Substituting \( d = 62.5 \) into \( A(d) \): \[ A(62.5) = 62.5(250 - 2 \cdot 62.5) = 62.5(250 - 125) = 62.5 \cdot 125 = 7812.5 \text{ ft}^2. \]
- Thus, the range of areas will go from \( 0 \) to \( 7812.5 \) square feet. A reasonable scale for the vertical axis would be in increments of 500 square feet. This results in a vertical scale from 0 to 8000 ft² for clarity.
Therefore, the complete statements are:
- A reasonable scale for the horizontal axis of the function \( A(d) = d(250 - 2d) \) is 0 to 130 feet, in increments of 10 feet.
- A reasonable scale for the vertical axis of the function \( A(d) = d(250 - 2d) \) is 0 to 8000 square feet, in increments of 500 square feet.
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