A solid, cone-shaped lead crystal paper weight has a height of 5 centimeters and a base diameter that is 20% larger than the height. If the density of lead crystal is `3.1g"/"cm^(3)`, what is the approximate mass of the paperweight? Round your answer to the nearest gram.

Answer

To find the mass of the cone-shaped lead crystal paperweight, we will follow these steps:

### Step 1: Determine the height and diameter of the cone
The height of the cone is given as 5 cm. The base diameter is 20% larger than the height.

**Calculation:**
\[ \text{Diameter} = \text{Height} + 0.2 \times \text{Height} = 1.2 \times \text{Height} = 1.2 \times 5 \, \text{cm} = 6 \, \text{cm} \]

### Step 2: Calculate the radius of the base
The radius is half of the diameter.

**Calculation:**
\[ \text{Radius} = \frac{\text{Diameter}}{2} = \frac{6 \, \text{cm}}{2} = 3 \, \text{cm} \]

### Step 3: Calculate the volume of the cone
The formula for the volume \( V \) of a cone is:
\[ V = \frac{1}{3} \pi r^2 h \]
Substituting the values of radius and height:
\[ V = \frac{1}{3} \pi (3 \, \text{cm})^2 (5 \, \text{cm}) = \frac{1}{3} \pi (9 \, \text{cm}^2) (5 \, \text{cm}) = \frac{1}{3} \pi (45 \, \text{cm}^3) \]

### Step 4: Simplify the volume
Calculating the volume:
\[ V = 15 \pi \, \text{cm}^3 \]

### Step 5: Calculate the mass using density
The mass \( m \) can be calculated using the formula:
\[ m = \text{Density} \times \text{Volume} \]
Given that the density of lead crystal is \( 3.1 \, \text{g/cm}^3 \):
\[ m = 3.1 \, \text{g/cm}^3 \times 15 \pi \, \text{cm}^3 \]

### Step 6: Calculate the numerical value
Using \( \pi \approx 3.14 \):
\[ m = 3.1 \times 15 \times 3.14 \approx 3.1 \times 47.1 \approx 146.01 \, \text{g} \]

### Step 7: Round the mass to the nearest gram
Rounding \( 146.01 \, \text{g} \) gives:
\[ m \approx 146 \, \text{g} \]

Thus, the approximate mass of the paperweight is **146 grams**.

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