Problem 21 - Calculus Early Transcendentals: Pearson New International Edition

Question

In determining the specific gravity of an object, its weight in air is found to be \(A=36\) pounds and its weight in water is \(W=20\) pounds, with a possible error in each measurement of \(0.02\) pound. Find, approximately, the maximum possible error in calculating its specific gravity \(S\), where \(S=A /(A-W)\).

Step-by-Step Solution

Answer

The maximum possible error in calculating the specific gravity is approximately 0.007.

Step 1: Formula for Specific Gravity

The specific gravity of the object is given by the formula \( S = \frac{A}{A - W} \), where \( A \) is the weight in air and \( W \) is the weight in water.

Step 2: Differentiate the Specific Gravity Formula

To find the maximum possible error in \( S \), first differentiate \( S \) with respect to \( A \) and \( W \). The derivative \( \frac{dS}{dA} \) is \( \frac{W}{(A-W)^2} \), and \( \frac{dS}{dW} \) is \( \frac{A}{(A-W)^2} \).

Step 3: Calculate the Error for Each Variable

The error in \( A \) is \( dA = 0.02 \) pounds, and the error in \( W \) is \( dW = 0.02 \) pounds. Substitute these values into the derived expressions for \( \frac{dS}{dA} \) and \( \frac{dS}{dW} \).

Step 4: Apply the Error Propagation Formula

Use the error propagation formula: \( dS = \left| \frac{dS}{dA} \right| \cdot dA + \left| \frac{dS}{dW} \right| \cdot dW \). Substitute \( \frac{dS}{dA} = \frac{20}{16^2} \) and \( \frac{dS}{dW} = \frac{36}{16^2} \) with \( dA = dW = 0.02 \).

Step 5: Compute the Maximum Possible Error

Calculate \( dS = \frac{20}{256} \times 0.02 + \frac{36}{256} \times 0.02 = 0.0025 + 0.0045 = 0.007 \); hence, the maximum possible error in calculating \( S \) is approximately \( 0.007 \).

Key Concepts

Error Propagation

Error propagation is a method used to calculate the uncertainty or error in a result that is computed from one or more measured values. In our example, we are trying to find the maximum possible error in the calculation of specific gravity when we know there are small errors in the weight measurements.

In this specific gravity problem, we have weights measured in air and water, each with a small error of 0.02 pounds. To find how these errors affect the final calculation, we use error propagation techniques. This involves partial differentiation of the formula with respect to each variable and then combining these results using the error propagation formula:

• The partial derivative with respect to the weight in air, denoted as \(\frac{dS}{dA}\), gives us insight into how changes in \(A\) impact \(S\).
• Similarly, \(\frac{dS}{dW}\) tells us how changes in \(W\) influence \(S\).

By substituting known error values and the derived partial derivatives, we estimate the total possible error in our calculated specific gravity value.

Differentiation

Differentiation is a mathematical tool used to analyze how a change in one quantity affects another. It's crucial in finding relationships between variables and calculating rates of change. In the context of error propagation for specific gravity, differentiation helps us understand how small errors in our measurements have a cumulative impact.

The key differentiation results used in this exercise are:

• For the weight in air \(A\), the derivative \(\frac{dS}{dA} = \frac{W}{(A - W)^2}\) indicates how errors in \(A\) modify the specific gravity \(S\).
• For the weight in water \(W\), the derivative \(\frac{dS}{dW} = \frac{A}{(A - W)^2}\) tells us about the impact of errors in \(W\).

With these derivatives, we determine how each measurement's error contributes to the uncertainty in \(S\). Differentiation here gives a precise, mathematical way to evaluate error impacts.

Weight Measurement

Weight measurement is the foundation upon which we build further calculations like specific gravity. Here, weights are taken both in air and water, which helps in determining the specific gravity of an object.

Reliable measurements are crucial, as inaccuracies or errors can propagate to affect derived calculations:

• We measured the weight in air as \(36\) pounds, while in water it was \(20\) pounds.
• Each of these measurements has a potential error of \(0.02\) pounds.

These initial weight measurements directly influence the specific gravity formula: \( S = \frac{A}{A - W} \). Even small errors in measurement are essential to consider because they affect the precision of our specific gravity calculation. Properly understanding these measurements ensures better precision and reliability of the final result.