Problem 117 - Algebra for College Students

Question

Determine the number of ounces a filled carton of the given size may contain for the given relative error. $$ \left|\frac{x-x_{t}}{x_{t}}\right|=\text { relative error in } x $$ 64 -oz carton; relative error no greater than 0.05.

Step-by-Step Solution

Answer

The filled carton may contain 60.8 to 67.2 ounces.

Step 1: Understand the formula

The formula given is \ \ \( \left| \frac{x - x_t}{x_t} \right| = \text{relative error in } x \). Here, \( x_t \) represents the true value, which is 64 ounces, and the relative error must be no greater than 0.05.

Step 2: Set up the inequality

Rewrite the formula as an inequality to represent that the relative error must be no greater than 0.05. \ \ \( \left| \frac{x - 64}{64} \right| \leq 0.05 \)

Step 3: Remove the absolute value

Since the absolute value inequality \( \left| \frac{x - 64}{64} \right| \leq 0.05 \) can be split into two separate inequalities, rewrite it as: \ \ \( -0.05 \leq \frac{x - 64}{64} \leq 0.05 \)

Step 4: Solve the inequalities

To find the range of \( x \), multiply all parts of the inequality by 64: \ \ -0.05 \times 64 \leq x - 64 \leq 0.05 \times 64 \ \ -3.2 \leq x - 64 \leq 3.2. Add 64 to all parts of the inequality to isolate \( x \): \ \ 60.8 \leq x \leq 67.2.

Step 5: Interpret the result

The filled carton may contain anywhere between 60.8 ounces and 67.2 ounces to ensure that the relative error is no greater than 0.05.

Key Concepts

Absolute Value Inequalities

Absolute value inequalities involve finding the range of values that satisfy a condition involving the absolute value of a variable. In the context of measurements, understanding these inequalities helps us determine acceptable values that lie within a specific tolerance range. The given inequality in this exercise is \( \left\| \frac{x - 64}{64} \right\| \leq 0.05 \). This states that the relative difference between the actual and true value should not exceed 0.05. Splitting the absolute value inequality into two parts is crucial: \( -0.05 \leq \frac{x - 64}{64} \leq 0.05 \). This approach helps in solving the inequality by removing the absolute value.

Algebraic Calculations

Algebraic calculations play a critical role in solving inequalities like the one in this exercise. Here, you start by setting up the inequality \( \left\| \frac{x - 64}{64} \right\| \leq 0.05 \), and then split it into two: \( -0.05 \leq \frac{x - 64}{64} \leq 0.05 \). To isolate \( x \), you multiply each part by 64, leading to: \ -3.2 \leq x - 64 \leq 3.2. Finally, adding 64 to all parts gives: \ 60.8 \leq x \leq 67.2. Breaking down each step is key to understanding how to manipulate and solve these inequalities accurately.

Relative Error

Relative error measures the discrepancy between the actual value and the true value, scaled by the true value. It's a crucial concept in measurements to gauge the accuracy and precision of an instrument or process. The formula given for relative error is \( \left\| \frac{x - x_t}{x_t} \right\| \leq \text{relative error in } x \). For this exercise, \( x_t \) is 64 ounces, and the relative error limit is 0.05. This means the measured value \( x \) must lie within 5% of the true value of 64 ounces, resulting in an acceptable range between 60.8 ounces and 67.2 ounces. This ensures the measurements are within the permissible error range.