The volume of sphere is $4851~cm^{3}$.
Find its diameter.

Question

The volume of sphere is $4851~cm^{3}$. 
Find its diameter.

Accepted Answer

The correct option is 3:
To solve this problem, we need to recall the formula for the volume of a sphere and the relationship between radius and diameter.

Key Concept: The volume of a sphere ($V$) is given by the formula:
$$V = \frac{4}{3} \pi r^3$$
where $r$ is the radius of the sphere.
Also, the diameter ($d$) of a sphere is twice its radius:
$$d = 2r$$

Given:
Volume of the sphere ($V$) = $4851\~cm^{3}$

Step-by-step explanation:1.  **Substitute the given volume into the formula:**
    $$4851 = \frac{4}{3} \pi r^3$$

2.  **Use the value of $\pi = \frac{22}{7}$:**
    $$4851 = \frac{4}{3} \times \frac{22}{7} \times r^3$$
    $$4851 = \frac{88}{21} r^3$$

3.  **Solve for $r^3$:**
    To isolate $r^3$, multiply both sides by $\frac{21}{88}$:
    $$r^3 = 4851 \times \frac{21}{88}$$

4.  **Perform the multiplication and division:**
    Divide 4851 by 11: $4851 \div 11 = 441$.
    Divide 88 by 11: $88 \div 11 = 8$.
    $$r^3 = 441 \times \frac{21}{8}$$
    Since $441 = 21^2$, we have:
    $$r^3 = 21^2 \times \frac{21}{8}$$
    $$r^3 = \frac{21^3}{8}$$

5.  **Solve for $r$ by taking the cube root of both sides:**
    $$r = \sqrt$$3$${\frac{21^3}{8}}$$
    $$r = \frac{21}{2}$$
    $$r = 10.5\~cm$$

6.  **Calculate the diameter:**
    Since $d = 2r$:
    $$d = 2 \times 10.5\~cm$$
    $$d = 21\~cm$$Therefore, the diameter of the sphere is 21 cm.

The incorrect option is 1: 42 cm. This would be the result if the radius was mistakenly taken as 21 cm and then multiplied by 2, or if the cube root of 8 in the denominator was not taken.
The incorrect option is 2: 10.5 cm. This value is the radius of the sphere, not the diameter. The question specifically asks for the diameter.
The incorrect option is 4: 84 cm. This value is obtained if the calculated diameter (21 cm) is mistakenly multiplied by 4, or if there was an error in calculating the cube root and then multiplying by 2.