https://r2.erweima.ai/mathbot/image/1725986361490mfub6rpv.jpg

Let's solve the two problems, numbered 57 and 58, from the image.

### Problem 57:
**Given:**
- A disk (thin right circular cylinder) with a circumference of 10 inches.
- You can tolerate an error of at most 0.02 inches in the circumference.

**Find:**
- The allowable error in the diameter measurement.

**Solution:**

The circumference \( C \) of a circle is related to the diameter \( D \) by the formula:
\[
C = \pi D
\]

Rearranging for \( D \), we get:
\[
D = \frac{C}{\pi}
\]

Now, consider the error in the circumference, \( \Delta C = 0.02 \) inches. The corresponding error in the diameter, \( \Delta D \), can be found by differentiating the formula for \( D \) with respect to \( C \):

\[
\Delta D = \frac{\Delta C}{\pi}
\]

Substitute the given values:

\[
\Delta D = \frac{0.02}{\pi} \approx \frac{0.02}{3.14} \approx 0.0064 \text{ inches}
\]

So, the allowable error in the diameter measurement is approximately **0.0064 inches**.

---

### Problem 58:
**Given:**
- Fahrenheit and Celsius temperatures are related by the formula: 
  \[
  C = \frac{5}{9}(F - 32)
  \]
- A solution must be kept at \( 50^\circ \text{C} \) with an error of at most 3% (or 1.5° C).
- You only have a Fahrenheit thermometer.

**Find:**
- The allowable error in the Fahrenheit reading.

**Solution:**

The error in the Celsius temperature is given as 1.5° C. Let's find the corresponding error in Fahrenheit, \( \Delta F \).

First, differentiate the formula for Celsius temperature with respect to Fahrenheit:

\[
\frac{dC}{dF} = \frac{5}{9}
\]

This represents the rate of change of Celsius with respect to Fahrenheit.

Now, the error in Fahrenheit, \( \Delta F \), can be found using the relation:

\[
\Delta C = \frac{5}{9} \Delta F
\]

Rearrange for \( \Delta F \):

\[
\Delta F = \frac{9}{5} \Delta C
\]

Substitute the value \( \Delta C = 1.5 \) degrees:

\[
\Delta F = \frac{9}{5} \times 1.5 = 2.7 \text{ degrees Fahrenheit}
\]

So, the allowable error on the Fahrenheit thermometer is **2.7 degrees Fahrenheit**.

---

Would you like further details on these solutions or have any other questions?

### Follow-Up Questions:
1. What is the general formula to convert an error from one unit to another?
2. How does the error tolerance affect manufacturing processes?
3. How can you reduce measurement errors in practical applications?
4. What are the implications of a 3% error in scientific experiments?
5. How do differential calculus concepts apply to error analysis?

**Tip:** Always double-check your units when converting between systems to avoid significant errors in calculations!

Solve two error analysis problems involving a disk's circumference with a tolerance of 0.02 inches and a Celsius to Fahrenheit conversion with an allowable error. Understand key concepts like differentiation, error propagation, and temperature conversion formulas.

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