Find Mass of a Gold Coin: Step-by-Step Solution

This question asks us to find the mass of a gold coin given its physical dimensions (radius and thickness) and the density of gold. To solve this, we need to first calculate the volume of the coin and then use the relationship between mass, density, and volume.

Understanding the Gold Coin's Shape and Properties

A gold coin is typically shaped like a cylinder. We are given the following information:

• Radius of the coin: $r = 2 \text{ cm}$
• Thickness of the coin (which is the height of the cylinder): $h = 0.05 \text{ cm}$
• Density of gold: $\rho = 19.3 \text{ gm/cm}^3$

We need to find the mass of the gold coin.

Calculating the Volume of the Gold Coin

The volume of a cylinder is calculated using the formula:

$V = \pi r^2 h$

Where:

• $V$ is the volume
• $\pi$ is a mathematical constant (approximately 3.14159)
• $r$ is the radius of the base
• $h$ is the height (thickness)

Let's plug in the given values for the gold coin:

$V = \pi \times (2 \text{ cm})^2 \times 0.05 \text{ cm}$

$V = \pi \times 4 \text{ cm}^2 \times 0.05 \text{ cm}$

$V = (4 \times 0.05) \pi \text{ cm}^3$

$V = 0.2 \pi \text{ cm}^3$

Using the approximate value of $\pi \approx 3.14159$, the volume is:

$V \approx 0.2 \times 3.14159 \text{ cm}^3$

$V \approx 0.628318 \text{ cm}^3$

Calculating the Mass of the Gold Coin

The relationship between mass, density, and volume is given by the formula:

$\text{Mass} = \text{Density} \times \text{Volume}$

We have the density of gold and the calculated volume of the gold coin. Now, we can find the mass:

$\text{Mass} = 19.3 \text{ gm/cm}^3 \times 0.628318 \text{ cm}^3$

$\text{Mass} \approx 12.13753 \text{ gm}$

Comparing with Options

Let's look at the given options for the mass of the gold coin:

• 12.1 gm
• 0.06 gm
• 6.1 gm
• 0.03 gm

Our calculated mass, approximately 12.13753 gm, is very close to 12.1 gm.

Conclusion

Based on the calculations using the dimensions and density of the gold coin, the mass is approximately 12.1 gm.

Revision Table: Gold Coin Mass Calculation

Additional Information: Density and Volume Concepts

Density: Density is a fundamental physical property of a substance that describes how much mass is contained in a given volume. It is defined as mass per unit volume. The formula is $\rho = \text{Mass} / \text{Volume}$. Different materials have different densities. For example, gold is much denser than water or aluminum.

Volume: Volume is the amount of three-dimensional space occupied by an object. The formula for volume depends on the shape of the object. For simple shapes like cylinders, spheres, and cubes, there are standard formulas. For irregular shapes, volume can be measured using methods like displacement.

In this problem, understanding that a coin is a cylinder and knowing the formula for the volume of a cylinder were key steps. We then applied the definition of density to find the mass.