Problem 57 - University Calculus: Early Transcendentals

Question

Tolerance The height and radius of a right circular cylinder are equal, so the cylinder's volume is \(V=\pi h^{3} .\) The volume is to be calculated with an error of no more than \(1 \%\) of the true value. Find approximately the greatest error that can be tolerated in the measurement of \(h,\) expressed as a percentage of \(h\)

Step-by-Step Solution

Answer

The greatest allowable error in \( h \) is approximately 0.33% of \( h \).

Step 1: Understand the Relationship

We need to find the derivative of the volume with respect to the height. Since the formula for the volume of the cylinder is \( V = \pi h^3 \), we identify \( V \) as a function of \( h \): \( V(h) = \pi h^3 \). The error in \( V \), denoted as \( \Delta V \), must not exceed 1% of \( V \).

Step 2: Calculate Derivative of Volume

Compute the derivative of \( V \) with respect to \( h \). We have: \( \frac{dV}{dh} = 3\pi h^2 \). This derivative helps us understand how the volume changes with small changes in height.

Step 3: Relate Volume Error to Height Error

We use the differential approximation: \( \Delta V \approx \frac{dV}{dh} \times \Delta h \). We want this volume error to be less than or equal to \( 0.01\times V \), i.e., \( \frac{dV}{dh} \times \Delta h \leq 0.01 \times \pi h^3 \).

Step 4: Solve for Height Error

Substitute \( \frac{dV}{dh} = 3\pi h^2 \) into the inequality: \( 3\pi h^2 \Delta h \leq 0.01 \pi h^3 \). Simplify to get: \( \Delta h \leq \frac{0.01 h^3}{3 h^2} \). Thus, \( \Delta h \leq \frac{0.01}{3} h \).

Step 5: Convert to Percentage

To express \( \Delta h \) as a percentage of \( h \), divide by \( h \) and multiply by 100: \( \frac{\Delta h}{h} \times 100 \leq \frac{0.01}{3} \times 100 \approx 0.3333\% \).

Key Concepts

Error Analysis in Calculus

Error analysis is a crucial part of calculus and mathematical computations, especially when measurements are involved, such as in the problem of calculating the volume of a cylinder with a specific tolerance. The objective is to assess how small errors in measurements can affect the result of a calculation. When you deal with measurements, slight inaccuracies or errors in the data can lead to deviations in outcomes.
For instance, in this problem, we want the volume error of the cylinder to remain within 1% of the actual volume. This means any error in height measurement must contribute to the volume error by at most 1%.
The systematic approach to solve this exercise involves:

• Identifying the relationship between the variables involved (volume and height).
• Calculating the error in volume for a given error in height.
• Ensuring that this error stays within the permissible limit.

This type of error analysis allows us to ensure that our measurements and calculations are precise and reliable.

Calculating Cylinder Volume

The volume of a cylinder is a fundamental concept in geometry and calculus, often used to apply differentiation techniques. The formula for the volume of a right circular cylinder where height and radius are the same is given by \( V = \pi h^3 \). This compact formula succinctly expresses how the volume of a cylinder scales with the cube of its height when radius and height are equal.
In problems like this, understanding the volume formula's dependency on height helps in predicting how alterations in height will affect the volume. For a small change in height, say \( \Delta h \), the volume experiences a corresponding change \( \Delta V \). Calculating this change accurately is essential, especially when precision matters, as in maintaining a volume error within a specified bound.
Learning to derive such formulas and apply them thoughtfully in practical contexts is a valuable skill in both academic and real-world scenarios.

Understanding Derivative Approximation

The concept of derivative approximation sits at the heart of many practical applications of calculus. By using derivatives, one can estimate the change in a function's output due to a small change in input. This principle is crucial when dealing with errors in measurements.
For our cylinder problem, the derivative \( \frac{dV}{dh} = 3\pi h^2 \) tells us the rate at which the volume changes concerning the height. Using this derivative, we express the related change in volume as \( \Delta V \approx \frac{dV}{dh} \times \Delta h \). This approximation allows us to map the potential error in height to its impact on volume, thereby understanding and controlling the error propagation.

• Approximation provides quick estimates which can guide more careful, exact calculations if needed.
• It offers a means to connect changes in one variable directly to another.

Grasping how derivative approximation works and applying it correctly will strengthen not only your calculus skills but also your problem-solving capabilities across various disciplines.