Problem 201 - Calculus Volume 3

Question

The radius \(r\) and height \(h\) of a right circular cylinder are measured with possible errors of \(4 \%\) and \(5 \%\) respectively. Approximate the maximum possible percentage error in measuring the volume (Recall that the percentage error is the ratio of the amount of error over the original amount. So, in this case, the percentage error in \(V\) is given by \(\frac{d V}{V}\).)

Step-by-Step Solution

Answer

The maximum percentage error is 13%.

Step 1: Determine the Formula for Volume

The volume of a cylinder, \( V \), is given by the formula: \( V = \pi r^2 h \). Here, \( r \) is the radius and \( h \) is the height of the cylinder.

Step 2: Derive the Differential of the Volume Formula

To find the differential \( dV \), we differentiate the volume formula with respect to its variables: \[ dV = \frac{\partial V}{\partial r} \cdot dr + \frac{\partial V}{\partial h} \cdot dh \] \[ = 2\pi rh \cdot dr + \pi r^2 \cdot dh \]

Step 3: Calculate the Relative Error in Volume

The relative error in volume is calculated as:\[ \frac{dV}{V} = \frac{2\pi rh \cdot dr + \pi r^2 \cdot dh}{\pi r^2 h} \] Simplifying this gives:\[ \frac{dV}{V} = \frac{2\cdot dr}{r} + \frac{dh}{h} \]

Step 4: Substitute Percentage Errors for \(dr\) and \(dh\)

Since the percentage error in \( r \) is \( 4\% \), we have \( \frac{dr}{r} = 0.04 \), and since the percentage error in \( h \) is \( 5\% \), we have \( \frac{dh}{h} = 0.05 \).

Step 5: Calculate Maximum Percentage Error

Substitute the values from Step 4 into the relative error formula from Step 3:\[ \frac{dV}{V} = 2(0.04) + 0.05 = 0.08 + 0.05 = 0.13 \] Thus, the maximum possible percentage error in measuring the volume is \( 13\% \).

Key Concepts

Differential Calculus

Differential calculus is all about studying how things change. It's a branch of mathematics that focuses on derivatives and differentials, which are tools that help us understand how a function changes as its input changes slightly. In practical terms, differential calculus allows us to compute the rate at which something happens. This is incredibly useful for determining error margins, like in the example with the cylinder's volume measurement.

In our exercise, we use differential calculus to find how small changes in the cylinder’s radius and height affect the overall volume. By differentiating the volume formula, we obtained an expression for the differential of volume. This tells us how sensitive the volume is to changes in the radius and height with the partial derivatives.

Percentage Error

The percentage error is a way to describe how much error is present when you measure something. It's useful in real-life situations where measurements are not perfectly accurate.

This is calculated by taking the error amount, dividing it by the true measurement, and converting it to a percentage. The result tells you how large the error is relative to the original measurement.

• For instance, a 4% error in measuring the radius means you're off by 4% of the actual radius value.
• This concept helps determine how accurate or precise a measurement really is, allowing us to make more informed decisions when working with these measurements.

Volume of Cylinder Calculation

To understand how measurement errors impact volume, it's essential to know how to calculate the volume of a cylinder. The formula for volume is:\[ V = \pi r^2 h \]where \( r \) is the radius and \( h \) is the height.

Calculating the exact volume involves substituting the radius and height into this equation. Since both measurements can have errors, the calculated volume might be inaccurate.

• Knowing this formula is crucial since it’s the baseline measurement that gets adjusted when accounting for potential errors.
• Adjusting both \( r \) and \( h \) in the formula should give us a rough estimate of how errors can impact the calculated volume of a cylinder.

Relative Error

Relative error is the ratio of the absolute error to the true value. It provides a measure of the uncertainty involved in measurements, expressed as a fraction. In the cylinder exercise, relative error in volume is calculated by:\[ \frac{dV}{V} \]This relationship helps determine how inaccuracies in measurements of radius and height translate into errors in computed volume.

More specifically, relative error informs us how significant the measurement errors are in proportion to the volume itself.

• The relative error can also help compare errors across different measurements to see which one is causing the most inconsistency.
• It's especially useful in fields like engineering, where precise measurements are crucial.

Mathematical Derivatives

A derivative is a mathematical way to show how a function is changing at any given point. It is a key concept in calculus and is used extensively in finding differentials, like in the example where changes in the radius and height affect the volume of a cylinder.

Mathematically, derivatives show up when we differentiate functions.

• For the cylinder's volume formula, the derivative with respect to "r" gives us how volume changes with changes in radius, while the one with "h" shows changes as height varies.
• This concept forms the backbone of our error analysis process, allowing us to see the impact of small adjustments in the input variables.
• Understanding derivatives enables us to make projections about what could happen if the input measurements are slightly off from what we assume they are.