Problem 7 - Basic Engineering Mathematics

Question

Find the volume and surface area of a sphere of diameter \(8 \mathrm{~cm}\).

Step-by-Step Solution

Answer

Volume \(\approx 268.08 \mathrm{~cm}^3\); Surface Area \(\approx 201.06 \mathrm{~cm}^2\).

Step 1: Find the Radius

The diameter of the sphere is provided as \(8 \mathrm{~cm}\). To find the radius, divide the diameter by 2: \( r = \frac{8}{2} = 4 \mathrm{~cm} \).

Step 2: Calculate the Volume of the Sphere

The formula for the volume of a sphere is \( V = \frac{4}{3} \pi r^3 \). Substitute \( r = 4 \mathrm{~cm} \) into the formula: \[ V = \frac{4}{3} \pi (4)^3 = \frac{4}{3} \pi (64) = \frac{256}{3} \pi \approx 268.08 \mathrm{~cm}^3 \].

Step 3: Calculate the Surface Area of the Sphere

The formula for the surface area of a sphere is \( A = 4 \pi r^2 \). Substitute \( r = 4 \mathrm{~cm} \) into the formula: \[ A = 4 \pi (4)^2 = 4 \pi (16) = 64 \pi \approx 201.06 \mathrm{~cm}^2 \].

Key Concepts

Volume of a Sphere

The volume of a sphere represents the amount of three-dimensional space it occupies. Imagine filling up a spherical balloon with water; the measure of that water would be the volume of the sphere. To calculate this, we use a formula derived from calculus:

• Formula: \( V = \frac{4}{3} \pi r^3 \)

Where \(V\) is the volume, \(\pi\) (pi) is approximately 3.14159, and \(r\) represents the radius of the sphere.

To find the volume, substitute the radius into the formula and solve for \(V\). For example, in the case of our previous exercise with a radius of \(4 \mathrm{~cm}\), we calculate:\[ V = \frac{4}{3} \pi (4)^3 = \frac{256}{3} \pi \approx 268.08 \mathrm{~cm}^3 \]Breaking it down:

• \(4\) cubed is the radius raised to the power of three: \(4^3 = 64\).
• Multiply by \(\pi\) to retain the mathematical constant.
• Finally, multiply by the fraction \(\frac{4}{3}\).

Understanding the process helps you visualize how much space a sphere can occupy!

Surface Area of a Sphere

The surface area of a sphere is the total area that the surface of the sphere covers. It's like finding out how much material you'd need to make a basketball. The formula used to find the surface area is straightforward:

• Formula: \( A = 4 \pi r^2 \)

Where \(A\) is the surface area and \(r\) is the radius. This formula essentially tells us how much two-dimensional space wraps around the sphere.

Let's take our exercise example with a radius of \(4 \mathrm{~cm}\):\[ A = 4 \pi (4)^2 = 64 \pi \approx 201.06 \mathrm{~cm}^2 \]Here's how it works:

• Square the radius first, so \(4^2 = 16\).
• Multiply by \(\pi\), preserving the mathematical constant.
• Finally, multiply by 4 to cover all sides of the sphere's surface.

When you calculate the surface area, you're determining how much space the outer layer of the sphere takes up.

Radius of a Sphere

The radius of a sphere is a fundamental concept in geometry. It is the distance from the very center of the sphere to any point on its outer surface. If you imagine cutting a sphere in half, the radius would be the length of the line from the center to the outer edge of the circle formed.

Understanding the radius is crucial because it serves as the key dimension for determining both the volume and surface area of a sphere. Given the diameter of a sphere, you can find the radius easily:

• The diameter is a line passing through the center, connecting two points on the sphere's surface.
• Radius is half the diameter: \( r = \frac{d}{2} \).

So, for a sphere with a diameter of \(8 \mathrm{~cm}\), the radius would be:\[ r = \frac{8}{2} = 4 \mathrm{~cm} \]This simple calculation allows you to plug the radius into formulas for volume and surface area. The radius is the backbone of sphere geometry, providing a foundation for unlocking more complex calculations.