When Jaya divided surface area of a sphere by its volume, she got `(2)/(3) cm` as answer. What is the radius of sphere ?

Options

A. 5.4 cm
B. 4.6 cm
C. 4.5 cm
D. 3.5 cm

Answer

To find the radius of the sphere given that the ratio of its surface area to its volume is \( \frac{2}{3} \) cm, we can follow these steps:

### Step 1: Write down the formulas for surface area and volume of a sphere.
The surface area \( A \) of a sphere is given by:
\[ A = 4\pi r^2 \]
The volume \( V \) of a sphere is given by:
\[ V = \frac{4}{3}\pi r^3 \]

### Step 2: Set up the equation based on the given ratio.
According to the problem, the ratio of the surface area to the volume is:
\[ \frac{A}{V} = \frac{2}{3} \]
Substituting the formulas for surface area and volume, we have:
\[ \frac{4\pi r^2}{\frac{4}{3}\pi r^3} = \frac{2}{3} \]

### Step 3: Simplify the equation.
We can cancel \( \pi \) from the numerator and denominator:
\[ \frac{4r^2}{\frac{4}{3}r^3} = \frac{2}{3} \]
Now, simplifying further:
\[ \frac{4r^2 \cdot 3}{4r^3} = \frac{2}{3} \]
This simplifies to:
\[ \frac{12r^2}{4r^3} = \frac{2}{3} \]
Now, cancel \( 4 \) from the numerator and denominator:
\[ \frac{3r^2}{r^3} = \frac{2}{3} \]
This simplifies to:
\[ \frac{3}{r} = \frac{2}{3} \]

### Step 4: Cross-multiply to solve for \( r \).
Cross-multiplying gives:
\[ 3 \cdot 3 = 2 \cdot r \]
This simplifies to:
\[ 9 = 2r \]

### Step 5: Solve for \( r \).
Dividing both sides by 2:
\[ r = \frac{9}{2} = 4.5 \text{ cm} \]

### Conclusion:
The radius of the sphere is \( 4.5 \) cm.

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