Question

1.6 A hockey puck, whose diameter is approximately 3 inches, is to be used to determine the value of \(\pi\) to three significant figures by carefully measuring its diameter and its circumference. For this calculation to be done properly, the measurements must be made to the nearest _____________. a) hundredth of a \(\mathrm{mm}\) c) \(\mathrm{mm}\) e) in b) tenth of a \(\mathrm{mm}\) d) \(\mathrm{cm}\)

Short answer

Answer: The level of accuracy needed is measuring the diameter and circumference to the nearest tenth of a millimeter.

Step-by-step solution

Understand the problem

By calculating \(\pi\), we use the formula \(\pi = \frac{C}{d}\), where \(C\) represents the circumference, and \(d\) represents the diameter of a circle (in this case, the hockey puck). We want to calculate the value of \(\pi\) to three significant figures to determine the accuracy we need for the measurements.

Determine the desired level of accuracy for \(\pi\)

Since we want the value of \(\pi\) to be accurate to three significant figures, we can use the approximation \(\pi \approx 3.14\). In our calculations, we want errors to be smaller than half of the last significant figure, i.e., smaller than 0.005.

Express diameter and circumference in inches and verify the given diameter value

The problem states that the diameter of the hockey puck is approximately 3 inches. We will use this value to express the circumference. For a circle, circumference \(C = \pi d\). For a 3-inch diameter, the approximate circumference would be \(C = 3.14 \times 3 \approx 9.42\) inches.

Calculate the required accuracy for measuring the diameter and circumference

To determine the level of accuracy in measurement required to have the value of \(\pi\) accurate to three significant figures, we need to make sure that the potential errors in measurement are smaller than 0.005. Let's assume that our measurements have an error of \(e_d\) for the diameter and \(e_C\) for the circumference. To hold the specified \(\pi\) accuracy, we need: \(\frac{e_C}{e_d} < 0.005\) \(\Rightarrow e_C < 0.005 \cdot e_d\) Given that we know the approximate diameter (3 inches) and circumference (9.42 inches) of the hockey puck, we can calculate the required errors in inches.

Convert errors to the given units

With the help of the equation we got in Step 4, we can now find possible values for \(e_d\) and \(e_C\) in different units. Note that there are 2.54 cm (25.4 mm) in one inch. a) hundredth of a mm: \(\Rightarrow e_d = \frac{1}{100} \cdot \frac{1}{10} \,\text{in} \approx 0.000393701 \,\text{in}\) \(\Rightarrow e_C = 0.005 \cdot e_d \approx 0.00001968504 \,\text{in}\) b) tenth of a mm: \(\Rightarrow e_d = \frac{1}{10} \cdot \frac{1}{10} \,\text{in} \approx 0.00393701 \,\text{in}\) \(\Rightarrow e_C = 0.005 \cdot e_d \approx 0.0001968504 \,\text{in}\) c) mm: \(\Rightarrow e_d = \frac{1}{10} \,\text{in} \approx 0.0393701 \,\text{in}\) \(\Rightarrow e_C = 0.005 \cdot e_d \approx 0.0001968504 \,\text{in}\) d) cm: \(\Rightarrow e_d = \frac{1}{2.54} \,\text{in} \approx 0.393701 \,\text{in}\) \(\Rightarrow e_C = 0.005 \cdot e_d \approx 0.001968504 \,\text{in}\) e) in: \(\Rightarrow e_d = 1 \,\text{in}\) \(\Rightarrow e_C = 0.005 \cdot e_d = 0.005 \,\text{in}\)

Compare required errors to available options

Now, we compare the required errors found in Step 5 and the options provided. We can observe that: a) hundredth of a mm: 0.000393701 in (diameter) and 0.00001968504 in (circumference), which is less than the allowed error; b) tenth of a mm: 0.00393701 in (diameter) and 0.0001968504 in (circumference), which is less than the allowed error; c) mm: 0.0393701 in (diameter) and 0.001968504 in (circumference), which is higher than the allowed error; d) cm: 0.393701 in (diameter) and 0.001968504 in (circumference), which is higher than the allowed error; e) in: 1 in (diameter) and 0.005 in (circumference), which is higher than the allowed error.

Choose the correct answer

From the analysis in Step 6, the best option would be option b) tenth of a mm. Since it provides a level of accuracy that is less than the allowed error, it would help in determining the value of \(\pi\) to three significant figures.

Key concepts

Pi calculation

Pi (\(\pi\)) is a crucial mathematical constant representing the ratio of a circle's circumference to its diameter. Calculating \(\pi\) accurately is essential in geometry and various real-world applications. In this exercise, we use the hockey puck's diameter and circumference for the calculation.

• The formula is \(\pi = \frac{C}{d}\), where \(C\) is the circumference, and \(d\) is the diameter.
• Accurate measurement of both metrics is necessary to ensure precision.

To find \(\pi\) to three significant figures, start with a close approximation like 3.14. This ensures minimal error, crucial when converting from physical measurements to mathematical results.

Significant figures

Significant figures indicate the precision of a measurement. They're vital in conveying how reliable a number is, especially in scientific calculations. When expressing \(\pi\) to three significant figures:

• Start with 3.14, which holds three figures of importance.
• The goal is to minimize error beyond 0.005, representing half of the smallest significant digit.

This precision ensures that calculations, especially in the real world, remain reliable and accurate. Maintaining the correct amount of significant figures keeps errors in check, enhancing the outcome's validity.

Unit conversion

Unit conversion is foundational when working with measurements, ensuring consistency and accuracy. For the hockey puck, converting between inches, millimeters, and centimeters is crucial.

• 1 inch equals 2.54 centimeters or 25.4 millimeters.
• Proper conversions align dimensions to ensure accurate results.

In calculations, converting units correctly aids in precise outcomes. This ensures that the evaluation of \(\pi\) is consistent across different measurement systems, facilitating international scientific communication.

Error analysis

Error analysis involves evaluating the potential impact of inaccuracies in measurements. It ensures that the final result remains trustworthy and within acceptable limits.

• For three significant figures, errors must be less than 0.005.
• Assess the error impact using \(e_C < 0.005 \, \times \, e_d\), where \(e_C\) is the circumference error and \(e_d\) is the diameter error.

Selecting the appropriate measurement precision, like a tenth of a mm, helps minimize errors, leading to a more accurate \(\pi\) calculation. Proper error analysis fosters confidence in scientific and mathematical conclusions.