To determine a reasonable scale for the horizontal and vertical axes of the function \( A(d) = d(450 - 2d) \), we first need to analyze the function more closely.
1. Finding the domain:
The variable \( d \) represents a quantity where \( A \) is likely to represent an area. Given that \( A(d) \) involves the expression \( d(450 - 2d) \), it makes sense to restrict \( d \) to non-negative values:
- Since \( 450 - 2d \) must also be non-negative for \( A(d) \) to be meaningful in the context, we set: \[ 450 - 2d \geq 0 \implies 2d \leq 450 \implies d \leq 225 \]
So, the reasonable domain for \( d \) is \( 0 \leq d \leq 225 \).
2. Evaluating \( A(d) \):
Next, we evaluate \( A(d) \) at key points to set the scale for the vertical axis:
- \( A(0) = 0(450 - 2 \times 0) = 0 \)
- \( A(225) = 225(450 - 2 \times 225) = 225(0) = 0 \)
Now, we need to find the maximum area:
To find the maximum area, we can take the derivative and set it to zero or complete the square.
The function is a downward-opening parabola. The vertex form gives max \( d\): \[ d = -\frac{b}{2a} = -\frac{450}{-4} = 112.5 \] Calculating \( A(112.5) \): \[ A(112.5) = 112.5(450 - 2 \times 112.5) = 112.5(450 - 225) = 112.5 \times 225 = 25312.5 \]
Now that we know the function ranges from \( 0 \) to \( 25312.5 \), we can decide on the vertical axis scale.
Summary of reasonable scales:
- Horizontal Axis (d): A reasonable scale would be from 0 to 225, with intervals that make sense for the context, such as increments of 25.
- Vertical Axis (A): A reasonable scale would be from 0 to approximately 26000, which can also be in increments that zero in on the maximum area, e.g., increments of 5000.
Thus, you could fill in:
- A reasonable scale for the horizontal axis of the function A(d) = d(450-2d) is (0 to 225).
- A reasonable scale for the vertical axis of the function A(d) = d(450-2d) is (0 to 26000).