Problem 98 - College Algebra and Calculus: An Applied Approach

Question

You stop at a self-service gas station to buy 15 gallons of 87 -octane gasoline at \(\$ 2.42\) a gallon. The pump scale is accurate to within one-tenth of a gallon. How much money might you have been undercharged or overcharged?

Step-by-Step Solution

Answer

Due to the inaccuracy of the pump, you might have been overcharged or undercharged by \$0.242.

Step 1: Determine the Uncertainty in Volume

Since the pump scale is accurate to within one-tenth of a gallon, this means that you could have been given anywhere from \(15 - 0.1 = 14.9\) gallons to \(15 + 0.1 = 15.1\) gallons.

Step 2: Calculate the Monetary Value of the Uncertainty

The cost per gallon is \(\$2.42\). Therefore, the cost of the extra or missing 0.1 gallon is \(0.1 \times 2.42 = \$0.242\). Therefore, you could have been undercharged or overcharged by this amount.

Step 3: Interpret the Result

This result means that due to the inaccuracy of the pump, either 0.242 USD too much may have been charged, or 0.242 USD too little may have been charged.

Key Concepts

Gasoline Cost Calculation

Understanding how to calculate the cost of gasoline is a vital skill in today's economy, where fuel prices frequently fluctuate. The essential formula to remember is quite simple: the total cost is the product of the amount of gasoline in gallons and the price per gallon.

For instance, if you pump 15 gallons of gasoline priced at \$2.42\/gallon, the total cost is calculated by multiplying the two:
\( 15 \text{ gallons} \times \$2.42\/\text{gallon} = \$36.30 \).

However, real-life scenarios often involve more complexity due to factors like taxes, discounts, or, as we will explore further, measurement uncertainty at the pump.

Measurement Uncertainty

Measurement uncertainty arises in nearly all equipment and procedures where measurement is involved. It reflects the lack of exactness inherent in any measurement process. In the context of a gasoline pump, uncertainties can result from mechanical errors, calibration issues, or environmental conditions affecting the equipment's performance.

To mitigate this, there's a tolerance level, which, for gasoline pumps, is often one-tenth of a gallon. This range has financial implications for consumers, as it dictates that they might receive slightly more or less gasoline than displayed, affecting the overall cost.
When you're told the pump scale is accurate within one-tenth of a gallon, this implies a possible deviation of \(\pm 0.1\) gallon on each transaction, which could lead to being undercharged or overcharged by the corresponding value of the gasoline for that amount, a concept emphasized in our original exercise.

Real World Math Application

The practical application of mathematical concepts to real-life scenarios, like purchasing gasoline, equips individuals with tools to make informed financial decisions. In our example, we perform a simple calculation to understand how pump inaccuracy impacts our wallet.

To assess potential overcharge or undercharge, you simply take the maximum potential error in volume and multiply it by the cost per gallon:
\( \text{Monetary uncertainty} = \pm(0.1 \text{ gallon} \times \$2.42\/\text{gallon}) = \pm \$0.242 \).

This direct application of multiplication and awareness of measurement uncertainty shows how math is embedded in everyday transactions. It prepares individuals to question and understand the variances on their receipts, leading to a more mathematically informed society.