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 specific gravity is approximately 0.0041.

Step 1: Understand the Formula for Specific Gravity

The formula for specific gravity \(S\) is given as \(S = \frac{A}{A - W}\), where \(A\) is the weight in air, and \(W\) is the weight in water. This formula applies to objects measured under these conditions.

Step 2: Apply Given Values

Substitute the values \(A = 36\) pounds and \(W = 20\) pounds into the specific gravity formula: \[S = \frac{36}{36 - 20} = \frac{36}{16} = 2.25\]

Step 3: Determine the Possible Error in Measurements

Both \(A\) and \(W\) have a possible error of \(0.02\) pounds. We need to consider how this affects the calculation of \(S\).

Step 4: Use Differentials to Approximate the Error

Let the small changes (or errors) in \(A\) and \(W\) be \(dA\) and \(dW\), with \(dA = 0.02\) and \(dW = 0.02\). We use differentials: \[dS = \left| \frac{\partial S}{\partial A} \right| dA + \left| \frac{\partial S}{\partial W} \right| dW\].

Step 5: Calculate Partial Derivatives

Compute \(\frac{\partial S}{\partial A}\) and \(\frac{\partial S}{\partial W}\): \[\frac{\partial S}{\partial A} = \frac{1}{A-W} = \frac{1}{16} \] \[\frac{\partial S}{\partial W} = -\frac{A}{(A-W)^2} = -\frac{36}{16^2} = -\frac{36}{256} = -\frac{9}{64} \].

Step 6: Calculate the Maximum Possible Error in Specific Gravity

Substitute \(dA = 0.02\) and \(dW = 0.02\) into the error formula: \[dS = \left( \frac{1}{16} \times 0.02 \right) + \left( \frac{9}{64} \times 0.02 \right)\] Calculate: \[dS = 0.00125 + 0.0028125 = 0.0040625\].

Step 7: Final Result

The maximum possible error in calculating the specific gravity is approximately \(0.0041\). The calculated specific gravity is \(2.25\), so the value should be stated with an error range, \(2.25 \pm 0.0041\).

Key Concepts

Differential Calculus

Differential calculus is a subfield of calculus concerned primarily with finding rates at which quantities change. This is usually represented by the derivative of a function. Derivatives help us understand how a function's output changes as its input changes.

In the context of solving for specific gravity, differentials are used for error analysis. This involves determining how small changes in the input values (measurements of weight) affect the output value (specific gravity).

To do this, we consider the formula for the specific gravity: \[ S = \frac{A}{A-W} \]Differential calculus allows us to approximate the effect on \( S \) when there are small errors in \( A \) and \( W \). By using differentials, we apply the concept of partial derivatives to analyze how sensitive \( S \) is to small changes in its variables.

Error Analysis

Error analysis involves quantifying the uncertainty in measurements and calculations. Here, knowing the possible errors in the measurements of weight is crucial to understanding how they propagate through the formula for specific gravity.

The error in calculation is linked closely to the sensitivity of the specific gravity to changes in its variables, \( A \) and \( W \). When both measurements have an error of \( 0.02 \) pounds, we need to estimate the total impact of these errors on the computed specific gravity.

By using the approximation \[ dS = \left| \frac{\partial S}{\partial A} \right| dA + \left| \frac{\partial S}{\partial W} \right| dW \]where \( dA \) and \( dW \) are small changes (errors) in \( A \) and \( W \), we quantify the maximum possible error in our result. This method is crucial for adjusting and interpreting computed results correctly, especially when precision is necessary.

Partial Derivatives

Partial derivatives are tools used in mathematics to find the rate of change of a multivariable function with respect to one of those variables while holding the others constant.

This concept is quite useful in calculating the error margin in specific gravity. It helps break down how sensitive the gravity is to each measurement independently.

Calculate the partial derivatives for our problem as follows:

• \( \frac{\partial S}{\partial A} = \frac{1}{A-W} \)
• \( \frac{\partial S}{\partial W} = -\frac{A}{(A-W)^2} \)

These tell us how a small change in \( A \) or \( W \) will impact \( S \). When these are applied to the specific gravity formula, they allow us to determine the contribution of each variable's uncertainty to the overall error. As shown, these derivatives are crucial for rigorously determining the maximum possible error in the context of specific gravity measurements, especially with known measurement inaccuracies.