Problem 57 - Calculus Early Transcendentals: Pearson New International Edition

Question

On a lathe, you are to turn out a disk (thin right circular cylinder) of circumference 10 inches. This is done by continually measuring the diameter as you make the disk smaller. How closely must you measure the diameter if you can tolerate an error of at most \(0.02\) inch in the circumference?

Step-by-Step Solution

Answer

The diameter must be measured to within approximately 0.0064 inches.

Step 1: Understand the relationship between diameter and circumference

Recall the relationship between the diameter \(d\) and the circumference \(C\) of a circle, which is given by the formula \(C = \pi d\). In this problem, the circumference is given as 10 inches.

Step 2: Express circumference error in terms of diameter

The error in circumference \(\Delta C\) is related to the error in the diameter \(\Delta d\) by differentiating the formula \(C = \pi d\), yielding \(\Delta C = \pi \Delta d\). We want \(\Delta C \leq 0.02\) inches.

Step 3: Solve for the allowable error in the diameter

Re-arrange \(\Delta C = \pi \Delta d\) to solve for \(\Delta d\), giving \(\Delta d = \frac{\Delta C}{\pi}\). Substituting \(\Delta C = 0.02\) gives \(\Delta d = \frac{0.02}{\pi}\) inches.

Step 4: Calculate the maximum allowable error in diameter

Calculate the value of \(\Delta d\) using \(\pi \approx 3.14159\). Therefore, \(\Delta d \approx \frac{0.02}{3.14159} \approx 0.00637\) inches. This is the precision needed for diameter measurement.

Key Concepts

Error Propagation

When conducting measurements, tiny errors can result in larger deviations from the expected outcomes if not managed properly. This is where error propagation comes into play. In our exercise, we see how changes or errors in one variable can impact another. For example, an error in the diameter measurement of the disk affects its circumference. Understanding how errors accumulate helps in managing and predicting the reliability of measurements.

To handle error propagation:

• Identify all variables involved. Here, it's the diameter and circumference.
• Recognize the relationship between them, such as through differentiation.
• Estimate the maximum allowable error for reliable outcomes.

Through careful calculations, such as differentiating the formula, you can make sure that any measurement errors stay within acceptable limits.

Differentiation

Differentiation is a powerful mathematical tool that allows us to understand how functions change and interact. In the context of this problem, we used differentiation to understand how a small change in the diameter of a disk would affect its circumference.

Differentiation steps involved in this problem include:

• Taking the known formula for circumference, \(C = \pi d\), and differentiating it with respect to the diameter \(d\).
• The differentiation gives \(\Delta C = \pi \Delta d\), a way to relate changes in circumference to changes in diameter.

By differentiating, we directly translate infinitesimal changes in one variable into the potential error in another. This relation helps us design precise manufacturing settings, such as setting the needed precision in diameter measurements to ensure the overall circumference error stays below 0.02 inches.

Geometric Measurements

In practical applications, geometric measurements often require a high degree of precision. Measuring the diameter of a geometric shape, like a disk in this exercise, demands careful consideration of potential errors.

The principles involved include:

• Understanding geometric formulas and their use in real life, such as \(C = \pi d\) for a circle.
• Calculating the real-world impacts of measurement errors using these formulas.
• Knowing how precise the measurement tools need to be to achieve the desired tolerance levels.

With geometric measurements, the focus is often on both the actual measurement and the impact of errors. In our scenario, ensuring that the diameter is measured to within approximately 0.00637 inches is critical to maintaining the desired 10-inch circumference with minimal error.