Calculating the Weight of a Hollow Spherical Shell

This solution explains the process to determine the weight of a hollow spherical shell using its material density and dimensions, expressed in kilograms.

Given Information:

• Material Density: $$\rho = 4 \text{ g/cm}^3$$
• Internal Radius: $$r_1 = 15 \text{ cm}$$
• External Radius: $$r_2 = 18 \text{ cm}$$
• Value of Pi: $$\pi = \frac{22}{7}$$
• Formula Provided: Density = $$\frac{\text{Mass}}{\text{Volume}}$$

Step 1: Calculate the Volume of the Hollow Spherical Shell

The volume of a hollow object like this shell is found by subtracting the volume of the inner space from the volume of the outer shape.

The formula for the volume of a single sphere is $$V_{sphere} = \frac{4}{3}\pi r^3$$.

For a hollow spherical shell, the volume ($$V_{shell}$$) is the volume of the outer sphere minus the volume of the inner sphere:

$$ V_{shell} = V_{outer} - V_{inner} $$

Using the formula for the volume of a sphere:

$$ V_{shell} = \frac{4}{3}\pi r_2^3 - \frac{4}{3}\pi r_1^3 $$

We can factor out the common term $$\frac{4}{3}\pi$$:

$$ V_{shell} = \frac{4}{3}\pi (r_2^3 - r_1^3) $$

Now, let's calculate the cubes of the radii:

$$ r_2^3 = 18^3 = 18 \times 18 \times 18 = 5832 \text{ cm}^3 $$

$$ r_1^3 = 15^3 = 15 \times 15 \times 15 = 3375 \text{ cm}^3 $$

Find the difference between the cubes:

$$ r_2^3 - r_1^3 = 5832 - 3375 = 2457 \text{ cm}^3 $$

Substitute these values back into the volume formula:

$$ V_{shell} = \frac{4}{3} \times \frac{22}{7} \times 2457 \text{ cm}^3 $$

To simplify the calculation, we can divide 2457 by 7:

$$ \frac{2457}{7} = 351 $$

So the expression becomes:

$$ V_{shell} = \frac{4}{3} \times 22 \times 351 \text{ cm}^3 $$

Now, divide 351 by 3:

$$ \frac{351}{3} = 117 $$

The volume calculation is now:

$$ V_{shell} = 4 \times 22 \times 117 \text{ cm}^3 $$

$$ V_{shell} = 88 \times 117 \text{ cm}^3 $$

$$ V_{shell} = 10296 \text{ cm}^3 $$

Step 2: Calculate the Mass of the Shell

We use the given formula: Mass = Density $$\times$$ Volume.

$$ \text{Mass} = \rho \times V_{shell} $$

Substitute the density and the calculated volume:

$$ \text{Mass} = 4 \text{ g/cm}^3 \times 10296 \text{ cm}^3 $$

Multiplying these values gives the mass in grams:

$$ \text{Mass} = 41184 \text{ g} $$

Step 3: Convert Mass from Grams to Kilograms

The question asks for the weight in kilograms (kg). We need to convert the mass from grams (g) to kilograms.

Recall the conversion factor: 1 kg = 1000 g.

To convert grams to kilograms, divide the mass in grams by 1000:

$$ \text{Weight (in kg)} = \frac{\text{Mass (in g)}}{1000 \text{ g/kg}} $$

$$ \text{Weight (in kg)} = \frac{41184}{1000} $$

$$ \text{Weight (in kg)} = 41.184 \text{ kg} $$

Final Result

The calculated weight of the hollow spherical shell is 41.184 kg.