Problem 58 - Prealgebra

Question

The Susan B. Anthony dollar shown here has a radius of 0.52 inches and a thickness of 0.0079 inches. Find the following, rounding your answers to the nearest ten thousandth, if necessary. Use 3.14 for \(\pi\). a. The circumference of the coin. b. The area of one face of the coin. c. The volume of the coin.

Step-by-Step Solution

Answer

a. 3.2624 inches; b. 0.8491 square inches; c. 0.0027 cubic inches.

Step 1: Finding the Circumference

To find the circumference of the coin, we'll use the formula for the circumference of a circle, which is \( C = 2 \pi r \), where \( r \) is the radius of the circle. In this problem, the radius \( r \) is 0.52 inches.Given the value of \( \pi \) is 3.14, the circumference can be calculated as:\[C = 2 \times 3.14 \times 0.52\]\[C = 3.2624 \text{ inches}\]

Step 2: Calculating the Area of One Face

The area of one face of the coin is calculated using the formula for the area of a circle, \( A = \pi r^2 \), where \( r \) is the radius.Using the given radius \( r = 0.52 \) inches and \( \pi = 3.14 \):\[A = 3.14 \times (0.52)^2\]\[A = 3.14 \times 0.2704\]\[A = 0.849056 \text{ square inches}\]

Step 3: Calculating the Volume of the Coin

To find the volume of the coin, we can think of the coin as a cylinder. The volume \( V \) of a cylinder is calculated using the formula \( V = \pi r^2 h \), where \( h \) is the height (or thickness) of the cylinder.Using \( r = 0.52 \) inches, \( h = 0.0079 \) inches, and \( \pi = 3.14 \):\[V = 3.14 \times (0.52)^2 \times 0.0079\]\[V = 3.14 \times 0.2704 \times 0.0079\]\[V = 0.0026793 \text{ cubic inches}\]

Key Concepts

Circumference Calculation

To calculate the circumference of a circle, you need to know the radius, as the formula is \( C = 2 \pi r \). The radius is the distance from the center of the circle to any point on its boundary. For the Susan B. Anthony dollar, the radius is 0.52 inches. By substituting the radius into the formula, we calculate the circumference. Thus, \( C = 2 \times 3.14 \times 0.52 = 3.2624 \) inches. This is the distance around the coin, similar to measuring the length of a piece of string that wraps around the circle.

• The radius (r) is crucial as it directly affects the circumference.
• \( \pi \approx 3.14 \) is a constant that relates the circumference of any circle to its diameter.
• The circumference is always a linear measurement, i.e., it is expressed in inches or centimeters, not square or cubic units.

Area of a Circle

The area of a circle is calculated using the formula \( A = \pi r^2 \). This formula tells you how much space is inside the circle. For the Susan B. Anthony dollar, using a radius of 0.52 inches, the area calculates as \( A = 3.14 \times (0.52)^2 \), which simplifies to \( A = 0.849056 \) square inches. Area is always expressed in square units because it represents a two-dimensional space.

• The radius is a vital component for finding the area because it is squared, meaning small changes in the radius lead to larger changes in the area.
• The \( \pi \) constant again is instrumental in relating dimensions of circles.
• The area provides a measure of how much material you would need to cover the face of the coin, assuming it’s perfectly flat.

Volume of a Cylinder

When calculating the volume of a coin-like shape, which can be viewed as a cylinder, you use the formula \( V = \pi r^2 h \), where \( h \) is the height or thickness of the cylinder. For our exercise, the radius is 0.52 inches and the thickness (height) is 0.0079 inches. Plugging into the formula, we have \( V = 3.14 \times (0.52)^2 \times 0.0079 \), which equals \( 0.0026793 \) cubic inches. Volume tells us how much three-dimensional space the coin occupies.

• The height or thickness is crucial in extending the area calculation into the third dimension.
• Volume is always expressed in cubic units because it adds a dimensional depth to the flat face of the circle.
• This calculation helps to understand not just the face area of the coin, but the total space it occupies.