Problem 56 - Thomas Calculus

Question

Estimate the allowable percentage error in measuring the diameter \(D\) of a sphere if the volume is to be calculated correctly to within 3\(\%\) .

Step-by-Step Solution

Answer

The allowable percentage error in measuring the diameter \(D\) is 1\(\%\).

Step 1: Understand the Volume Formula

The volume of a sphere is given by the formula \( V = \frac{4}{3}\pi D^3 \). Here, \(D\) is the diameter of the sphere. Our goal is to calculate the allowable percentage error in \(D\) given that the volume should be accurate within 3\(\%\).

Step 2: Relate Changes in Diameter and Volume

First, find the relationship between small changes in diameter, \(dD\), and the resulting changes in volume, \(dV\). The derivative of the volume with respect to the diameter is \( \frac{dV}{dD} = 4\pi D^2 \). This can be used to estimate the change in volume: \(dV \approx \frac{dV}{dD} \cdot dD = 4\pi D^2 \cdot dD\).

Step 3: Calculate Percent Error in Volume

The allowable error in volume \(dV\) is a fraction of the calculated volume: \(dV = 0.03V = 0.03 \times \frac{4}{3}\pi D^3 \). Compare this with the expression from Step 2.

Step 4: Equate Error Expressions

Set the expression for \(dV\) found in Step 2 equal to the allowable error: \(4\pi D^2 \cdot dD = 0.03 \times \frac{4}{3}\pi D^3 \). Simplify to find \(dD\) as a fraction of \(D\).

Step 5: Simplify and Solve for Allowable Error in Diameter

From the equation in Step 4, divide both sides by \(4\pi D^2\), obtaining \(dD = 0.01D\). This implies that the allowable percentage error in \(D\) is 1\(\%\).

Key Concepts

Volume of a Sphere

To understand the volume of a sphere, consider it as a perfectly round three-dimensional object. The mathematical formula for calculating the volume of a sphere is given by:

• Formula: \( V = \frac{4}{3}\pi D^3 \)

Here, \(D\) represents the diameter of the sphere, not the radius. This is crucial because most formulas often use the radius.
The diameter is simply twice the radius \( (D = 2r) \). This formula helps us determine the volume by plugging in the diameter's value.
It's important in practical applications, like determining how much material is needed to fill a spherical container or how much space a sphere occupies.

Differentiation

Differentiation is a key concept in calculus that helps us understand how a function changes when its input changes.
When dealing with real-world physical attributes, differentiation helps us calculate small changes in measurements. In the context of a sphere’s volume with respect to its diameter, we write:

• Derivative of Volume: \( \frac{dV}{dD} = 4\pi D^2 \)

This derivative tells us how much the volume changes (\(dV\)) for a small change in the diameter (\(dD\)).
Understanding this relationship is essential to error estimation, as it allows us to connect errors in one measurement (diameter) to another (volume).

Percentage Error

Percentage error allows us to understand how significant an error is relative to the actual value.
In our problem, the volume of the sphere must be accurate within 3% of its value. This means any error in volume measurement shouldn’t exceed 3%.
To move from volume to diameter error, we equate the error caused by diameter change to this permissible volume error:

• Allowable Volume Error: \( dV = 0.03 \times V \)
• Allowable Diameter Error: \( dD = 0.01 \times D \)

This shows how the percentage error in the diameter directly influences the error in the volume.

Geometric Measurements

Geometric measurements form the foundation for accurate and reliable calculations across numerous applications. When measuring dimensions like the diameter of a sphere, even the slightest variance can have a large effect on other computed values, like volume.
In practical contexts, precision in measuring instruments and accurate calculations are paramount. Errors in measurement necessitate compensatory calculations like those explored with calculus to ensure that any deviations remain controlled and acceptable for the intended use.
This is why understanding calculus error estimation is critical in balancing practicality with precision, ensuring the success of applications ranging from simple geometrical tasks to complex engineering projects.