Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Question
![**Problem Statement:**
Use differentials to estimate the amount of paint needed to apply a coat of paint 0.05 cm thick to a hemispherical dome with a diameter of 50 meters.
**Solution:**
1. Convert thickness from centimeters to meters:
- \(0.05 \, \text{cm} = 0.0005 \, \text{m}\).
2. Determine the radius of the hemisphere:
- Radius \( r = \frac{50 \, \text{m}}{2} = 25 \, \text{m}\).
3. Use the formula for the surface area of a hemisphere:
- Surface area \( A = 2\pi r^2 \).
4. Calculate the differential of the volume:
- The differential \( dV = A \times \text{thickness} \).
5. Substitute the values:
- \( A = 2\pi (25)^2 \).
- \( dV = 2\pi (25)^2 \times 0.0005 \).
6. Solve for \( dV \) to estimate the volume of paint needed in cubic meters.
**Answer:**
\[
\boxed{} \, m^3
\]
**Explanation:**
The problem involves using calculus to estimate how much paint is needed to cover a hemispherical surface, using the concept of differentials to approximate the change in volume. The conversion to meters is crucial for maintaining consistency in units, and the problem emphasizes the application of differential calculus in practical scenarios.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F341515a7-9f2b-49dc-a023-366552e2ebc5%2F49d98d33-2f17-49a4-b686-5847eb21b862%2Fd3x9869_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
Use differentials to estimate the amount of paint needed to apply a coat of paint 0.05 cm thick to a hemispherical dome with a diameter of 50 meters.
**Solution:**
1. Convert thickness from centimeters to meters:
- \(0.05 \, \text{cm} = 0.0005 \, \text{m}\).
2. Determine the radius of the hemisphere:
- Radius \( r = \frac{50 \, \text{m}}{2} = 25 \, \text{m}\).
3. Use the formula for the surface area of a hemisphere:
- Surface area \( A = 2\pi r^2 \).
4. Calculate the differential of the volume:
- The differential \( dV = A \times \text{thickness} \).
5. Substitute the values:
- \( A = 2\pi (25)^2 \).
- \( dV = 2\pi (25)^2 \times 0.0005 \).
6. Solve for \( dV \) to estimate the volume of paint needed in cubic meters.
**Answer:**
\[
\boxed{} \, m^3
\]
**Explanation:**
The problem involves using calculus to estimate how much paint is needed to cover a hemispherical surface, using the concept of differentials to approximate the change in volume. The conversion to meters is crucial for maintaining consistency in units, and the problem emphasizes the application of differential calculus in practical scenarios.
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