Problem 52 - College Physics

Question

As you eat your way through a bag of chocolate chip cookies, you observe that each cookie is a circular disk with a diameter of 8.50 \(\mathrm{cm}\) and a thickness of 0.050 \(\mathrm{cm} .\) Find (a) the volume of a single cookie and (b) the ratio of the diameter to the thickness, and express both in the proper number of significant figures.

Step-by-Step Solution

Answer

(a) 8.92 cm³; (b) 170

Step 1: Calculate the Radius

To find the volume of a cookie, first calculate the radius from the diameter. Given the diameter is 8.50 cm, divide by 2 to get the radius. \[ \text{Radius} = \frac{8.50 \text{ cm}}{2} = 4.25 \text{ cm} \]

Step 2: Use the Volume Formula for a Cylinder

The volume of a cylinder is given by the formula \( V = \pi r^2 h \), where \( r \) is the radius and \( h \) is the thickness of the cookie. Here, use \( r = 4.25 \text{ cm} \) and \( h = 0.050 \text{ cm} \).\[ V = \pi (4.25 \text{ cm})^2 (0.050 \text{ cm}) \]Calculate:\[ V = \pi (18.0625 \text{ cm}^2) (0.050 \text{ cm}) \]\[ V = 2.8410625\pi \text{ cm}^3 \]Using \( \pi \approx 3.14159 \):\[ V \approx 8.922 \text{ cm}^3 \]

Step 3: Round the Volume to Significant Figures

Since the diameter is given to three significant figures (8.50 cm), the final answer for the volume should also have three significant figures. Thus, the volume is:\[ V \approx 8.92 \text{ cm}^3 \]

Step 4: Calculate the Ratio of Diameter to Thickness

Compute the ratio by dividing the diameter by the thickness:\[ \text{Ratio} = \frac{\text{Diameter}}{\text{Thickness}} = \frac{8.50 \text{ cm}}{0.050 \text{ cm}} \]\[ \text{Ratio} = 170 \]

Step 5: Ensure Proper Significant Figures for the Ratio

Both the diameter and the thickness are given to three significant figures, so the ratio should also be expressed with three significant figures: \[ 170 \]

Key Concepts

Significant Figures

Significant figures are crucial when presenting measurements and calculations because they reflect the precision of the original measurement. They tell us how many digits in a number are reliable and meaningful. For example, if the diameter of a cookie is given as 8.50 cm with three significant figures, it implies that our measurement is accurate to the hundredth of a centimeter.

When performing operations like multiplication or division, the result should be rounded to the same number of significant figures as the least precise measurement used. In the example of the cookie volume, both the diameter and thickness dictate that our answer should be rounded to three significant figures. Thus, 8.922 cm³ is presented as 8.92 cm³. This ensures that our presentation of the result accurately reflects the precision of the measurements used.

Cylinder Formula

The volume of a cylinder is calculated using the formula \[ V = \pi r^2 h \] where \( r \) is the radius and \( h \) is the height or thickness in this context. For our cookie, its shape can be modeled as a cylinder with a thickness of 0.050 cm and a diameter where the radius, \( r \), is half of 8.50 cm, translating to 4.25 cm.

Substituting these values into our cylinder formula, we calculate:

• \( V = \pi (4.25)^2 (0.050) \)
• Simplifying inside the formula gives us \( V = \pi (18.0625)(0.050) \)
• Finally multiplying by \( \pi \approx 3.14159 \) gives around 8.922 cm³.

This methodology reflects how geometric modeling can transform physical shapes into easy-to-calculate volumes using standard formulas.

Ratio Calculation

A ratio represents a comparison between two numbers, showing how many times one value contains or is contained within the other. For the cookie, calculating the ratio of its diameter to its thickness shows how wide the cookie is relative to its height.

The calculation proceeds by dividing the cookie's diameter by its thickness:

• Using the given values: \( \frac{8.50 \text{ cm}}{0.050 \text{ cm}} \)
• Simplifying this yields a ratio of 170.

Presenting this ratio also requires adhering to the rules of significant figures. Thus, both the measurements used must dictate the precision offered by our result. Here, the numbers given originally have three significant figures, yet because the outcome is a whole number in this simplified fraction, it retains its simplicity as 170, without needing additional zeros which signify excessive precision.

Geometry

Geometry assists in understanding the shapes and their properties around us. In this problem, the cookie is interpreted as a disk, a fundamental geometric shape characterized by a circular base and certain thickness, resembling the 3D shape known as a cylinder.

This geometric conceptualization, coupled with the use of measurements like diameter and thickness, allows us to apply the cylinder formula to find the volume. Understanding how these geometric properties translate to mathematical expressions and calculations facilitates the study of real-world objects through mathematics:

• The circle’s radius, halve of the diameter, helps us calculate the area of the base, \( \pi r^2 \).
• Multiplying this area with the height (thickness in this case) gives the volume.

These relationships underscore basic geometric principles essential for measuring and analyzing objects within various scientific and practical contexts.