Problem 57 - Prealgebra

Question

The \(\$ 1\) coin shown here depicts Sacagawea and her infant son. The diameter of the coin is \(26.5 \mathrm{mm},\) and the thickness is \(2.00 \mathrm{mm}\). Find the following, rounding your answers to the nearest hundredth. 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) 83.21 mm, b) 551.86 mm², c) 1103.73 mm³.

Step 1: Find the Circumference

The circumference of a circle is given by the formula \( C = \pi \times d \), where \( d \) is the diameter. For this coin, the diameter \( d = 26.5 \) mm.\[C = 3.14 \times 26.5 = 83.21 \text{ mm}\]Thus, the circumference of the coin is 83.21 mm.

Step 2: Calculate the Area of One Face

The area of a circle is given by \( A = \pi r^2 \), where \( r \) is the radius. First, find the radius by halving the diameter: \( r = 13.25 \) mm.\[A = 3.14 \times (13.25)^2 = 3.14 \times 175.5625 = 551.86 \text{ mm}^2\]Therefore, the area of one face of the coin is 551.86 mm².

Step 3: Determine the Volume

The volume of a cylinder is given by \( V = \pi r^2 h \), where \( h \) is the thickness. Using \( r = 13.25 \) mm and \( h = 2.00 \) mm:\[V = 3.14 \times (13.25)^2 \times 2.00 = 3.14 \times 175.5625 \times 2.00 = 1103.73 \text{ mm}^3\]So, the volume of the coin is 1103.73 mm³.

Key Concepts

Circumference

The circumference is essentially the distance around a circle. Think of it as the circle's outline. To find the circumference, you can use the formula:

• \( C = \pi \times d \)
• \( d \) is the diameter, which is the length of a line passing through the circle's center, connecting two points on its edge.

For the coin, the diameter is given as 26.5 mm. Plugging this into the formula:\[C = 3.14 \times 26.5 = 83.21\, \text{mm}\]Thus, the circumference of the Sacagawea dollar coin is 83.21 mm.

Area of a Circle

The area of a circle refers to the space enclosed within its boundary. Imagine it as how much space the circle covers. The formula to find it is:

• \( A = \pi r^2 \)
• \( r \) is the radius, half the diameter of the circle.

To find the radius, divide the diameter by 2:

• \( r = \frac{26.5}{2} = 13.25 \) mm

Then, substitute \( r \) into the area formula:\[A = 3.14 \times (13.25)^2 = 551.86\, \text{mm}^2\]This calculation tells us that the area of one face of the coin is 551.86 mm². Remember, area signifies the entire surface one side of the circle covers.

Volume of a Cylinder

Volume is the measurement of how much space an object occupies. For a cylindrical shape like a coin, the volume can be visualized as the amount of space inside this 3D shape. The formula for finding the volume of a cylinder is:

• \( V = \pi r^2 h \)
• \( h \) is the height or thickness of the cylinder.

We already have the radius from our earlier area calculation: 13.25 mm. The thickness \( h \) is given as 2.00 mm in this case. Thus, we substitute these values into the volume formula:\[V = 3.14 \times (13.25)^2 \times 2.00 = 1103.73\, \text{mm}^3\]This value indicates that the coin has a volume of 1103.73 mm³, representing the 3D space it occupies.