Problem 97 - College Algebra and Calculus: An Applied Approach

Question

You buy six T-bone steaks that cost \(\$ 7.99\) per pound. The weight listed on the package is \(5.72\) pounds. The scale that weighed the package is accurate to within \(\frac{1}{2}\) ounce. How much money might you have been undercharged or overcharged?

Step-by-Step Solution

Answer

You could have been overcharged or undercharged by approximately $0.25 due to the weight measurement's error tolerance.

Step 1: Weight Conversion

As the weight measurement's error tolerance is given in ounces, it should first be converted into pounds since the cost of the steak is in $/lb. Note there are 16 ounces in a pound. Therefore, \(\frac{1}{2}\) ounce should be converted to pounds: \(\frac{1}{2} \div 16 = 0.03125\) pounds.

Step 2: Calculate Potential Overcharge/Undercharge

Now, knowing that 0.03125 lbs could be the possible inaccurancy, calculate the potential overcharge or undercharge by multiplying this possible error in weight by the price per pound. Overcharge: \(0.03125 \times 7.99 = 0.25\$ \). Undercharge: \(-0.03125 \times 7.99 = -0.25\$ \).

Step 3: Interpret the result

Given the weight's error tolerance, there is a possibility of being overcharged or undercharged by $0.25. This means, you could have paid $0.25 too much if the steak weighed less than stated, or you could have gotten it $0.25 cheaper if it actually weighed more.

Key Concepts

Understanding Unit Conversion

When it comes to purchasing items like meat, it is important to ensure the weights are accurate, especially when prices are given per pound. However, sometimes the weight measurements might come with a certain error margin, such as in ounces, creating a need for unit conversion to maintain accuracy.

Converting units helps standardize the measure, making calculations straightforward. In this example, the scale's precision is within \( \frac{1}{2}\) ounce, so it's necessary to convert that to pounds since the price is in dollars per pound. With 16 ounces in a pound, performing the conversion \( \frac{1}{2} \div 16 = 0.03125\) pounds ensures consistency in calculations.

Identifying Potential Overcharge

Potential overcharge occurs when the listed weight might be higher than the actual weight. Consumers could end up paying more than necessary if the scale's error favors overestimation.

In this scenario, if the steaks were weighed inaccurately, you might pay for an additional \(0.03125\) pounds, leading to paying \(0.03125 \times 7.99 = 0.25\$\) more. Thus, understanding the effect of measurement errors can help ensure you’re charged fairly. It highlights the importance of knowing possible inaccuracies when buying goods by weight.

Mastering Cost Calculation

Cost calculation involves determining the total amount paid based on price per unit and the quantity or weight purchased. It's a fundamental concept in shopping and financial planning.

For the steaks priced at \(7.99\\(\) per pound, with a recorded weight of \(5.72\) pounds, the initial cost is calculated as \(7.99 \times 5.72 = 45.71\\)\). However, there might be discrepancies if the scale isn’t perfectly accurate. Understanding this process helps in comprehensively evaluating the correctness of the price charged at the checkout.

Evaluating Error Tolerance

Error tolerance refers to the permissible limit of error in measurement, which in this case, is \( \frac{1}{2}\) ounce. It's key in assessing how precise measurements need to be, especially when dealing with costly items or bulk purchases.

Considering a measurement error of 0.03125 pounds in either direction, you can be undercharged or overcharged by \(0.25\$\). Recognizing this potential variation can prepare consumers in questioning or verifying the weight details of priced commodities, ensuring financial fairness.