A hollow metal sphere has an external diameter of 12 cm and a thickness of 2 cm.
(i) Given that the mass of 1 cm' of the metal is
5.4 g, find the mass of the hollow sphere in kg.

To find the mass of the hollow metal sphere, we need to follow these steps:

1. Calculate the external radius of the sphere.
2. Calculate the internal radius of the sphere.
3. Calculate the volume of the metal used in the hollow sphere.
4. Calculate the mass of the hollow sphere using the given density of the metal.

**Step 1: Calculate the external radius of the sphere.**

The external diameter is 12 cm, so the external radius \( R \) is:
\[ R = \frac{12 \text{ cm}}{2} = 6 \text{ cm} \]

**Step 2: Calculate the internal radius of the sphere.**

The thickness of the sphere is 2 cm, so the internal radius \( r \) is:
\[ r = 6 \text{ cm} - 2 \text{ cm} = 4 \text{ cm} \]

**Step 3: Calculate the volume of the metal used in the hollow sphere.**

The volume of a sphere is given by the formula:
\[ V = \frac{4}{3} \pi r^3 \]

We need to find the volume of the metal, which is the difference between the volume of the external sphere and the volume of the internal sphere.

Volume of the external sphere (\( V_{\text{external}} \)):
\[ V_{\text{external}} = \frac{4}{3} \pi R^3 = \frac{4}{3} \pi (6)^3 = \frac{4}{3} \pi (216) = 288 \pi \text{ cm}^3 \]

Volume of the internal sphere (\( V_{\text{internal}} \)):
\[ V_{\text{internal}} = \frac{4}{3} \pi r^3 = \frac{4}{3} \pi (4)^3 = \frac{4}{3} \pi (64) = 85.33 \pi \text{ cm}^3 \]

Volume of the metal (\( V_{\text{metal}} \)):
\[ V_{\text{metal}} = V_{\text{external}} - V_{\text{internal}} = 288 \pi - 85.33 \pi = 202.67 \pi \text{ cm}^3 \]

**Step 4: Calculate the mass of the hollow sphere.**

We are given that the mass of 1 cm³ of the metal is 5.4 g.

First, find the volume of the metal in cm³:
\[ V_{\text{metal}} = 202.67 \pi \text{ cm}^3 \approx 202.67 \times 3.14159 = 636.57 \text{ cm}^3 \]

Now, calculate the mass in grams:
\[ \text{Mass} = \text{Volume} \times \text{Density} = 636.57 \text{ cm}^3 \times 5.4 \text{ g/cm}^3 = 3437.48 \text{ g} \]

Convert the mass to kilograms:
\[ \text{Mass} = 3437.48 \text{ g} \times \frac{1 \text{ kg}}{1000 \text{ g}} = 3.43748 \text{ kg} \]

Therefore, the mass of the hollow sphere is approximately:
\[ 3.437 \text{ kg} \]

Would you like more details or have any questions?

Here are some related questions for further exploration:

1. How would the mass change if the thickness were doubled?
2. What is the volume of the internal sphere?
3. How does the density of the metal affect the mass calculation?
4. What would be the mass if the external diameter were 10 cm?
5. How can the volume of a hollow object be experimentally determined?
6. What are the practical applications of hollow metal spheres in engineering?
7. How would the mass change if the density of the metal were 7 g/cm³?
8. How can we calculate the surface area of the hollow sphere?

**Tip:** When dealing with spherical objects, always remember to use the radius (half of the diameter) for volume and surface area calculations.

Learn how to calculate the mass of a hollow metal sphere with an external diameter of 12 cm and a thickness of 2 cm. This problem involves calculating volumes of spheres, using density formulas, and converting units. Suitable for Grades 9-12 students studying geometry and density calculations.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation