To determine how closely the machinist must control the radius, given an error tolerance of ±9 cm² in the area of the disk, we'll need to use the formula for the area of a circle:

$A = \pi r^2$

Where:

• $A$ is the area of the disk,
• $r$ is the radius of the disk,
• $\pi \approx 3.1416$.

Step-by-Step Approach:

1. Ideal Area: Suppose the ideal area of the disk is $A_0$.

2. Error Bound: The area of the disk can vary by ±9 cm², meaning the actual area $A$ can range between $A_0 - 9$ and $A_0 + 9$.

3. Find the range for the radius: To find how close the machinist must control the radius, solve for the radius corresponding to $A_0 - 9$ and $A_0 + 9$.

Let’s express the relationship between the area and the radius:

$A_0 = \pi r_0^2 \quad \text{(Ideal area)}$

The error-tolerant area bounds are:

$A_0 - 9 = \pi r_{\text{min}}^2$ $A_0 + 9 = \pi r_{\text{max}}^2$

Steps to solve:

• First, solve for $r_{\text{min}}$ and $r_{\text{max}}$: $r_{\text{min}} = \sqrt{\frac{A_0 - 9}{\pi}}$ $r_{\text{max}} = \sqrt{\frac{A_0 + 9}{\pi}}$

• The difference between $r_0$ and these bounds will tell how closely the machinist needs to control the radius.

To proceed, I need the value of the ideal area $A_0$. Could you provide that, or would you like me to assume a value to illustrate the process?