Question

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 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)**.