Question

The difference between volume and surface area of a sphere of radius of 10.5 cm is:

• A. 1486
• B. 4851
• C. 3465
• D. 3654

Hint:

Converting decimals (like $10.5$) to improper fractions (like $21/2$) facilitates rapid cancellation with the denominator of $\pi$ ($7$), minimizing complex multiplications.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
A sphere is a perfectly symmetrical three-dimensional geometric object.
We need to calculate both its Volume ($V$) and its Total Surface Area ($A$) using its radius $r$, and then find their absolute difference.
Key Formula or Approach:
The formula for the volume of a sphere is: \[ V = \frac{4}{3}\pi r^3 \] The formula for the total surface area of a sphere is: \[ A = 4\pi r^2 \] The difference is defined as: \[ \text{Difference} = |V - A| \] We will use $\pi \approx \frac{22}{7}$ and express the radius $10.5 \text{ cm}$ as a fraction: \[ r = 10.5 = \frac{21}{2} \text{ cm} \]

Step 2: Detailed Explanation:
First, let's calculate the Volume ($V$): \[ V = \frac{4}{3} \cdot \frac{22}{7} \cdot \left(\frac{21}{2}\right)^3 \] \[ V = \frac{4}{3} \cdot \frac{22}{7} \cdot \frac{21 \cdot 21 \cdot 21}{8} \] Simplify the constant coefficients: \[ V = \frac{4 \cdot 22}{21 \cdot 8} \cdot 21 \cdot 21 \cdot 21 \] Since $3 \cdot 7 = 21$: \[ V = \frac{88}{8 \cdot 21} \cdot 21 \cdot 441 = 11 \cdot 441 = 4851 \text{ cm}^3 \] Next, let's calculate the Surface Area ($A$): \[ A = 4 \cdot \frac{22}{7} \cdot \left(\frac{21}{2}\right)^2 \] \[ A = 4 \cdot \frac{22}{7} \cdot \frac{441}{4} \] Simplify by canceling the factor of 4: \[ A = \frac{22 \cdot 441}{7} \] Since $\frac{441}{7} = 63$: \[ A = 22 \cdot 63 = 1386 \text{ cm}^2 \] Now, find the difference between Volume and Surface Area: \[ \text{Difference} = V - A = 4851 - 1386 = 3465 \]

Step 3: Final Answer:
The difference between the volume and surface area of the sphere is 3465.