Which of the following is the ratio of the surface area of the sphere with radius r to its volume?

Options

A. `4/pi`
B. `3/r`
C. `r/4`
D. `r/pi`

Explanation

To find the ratio of the surface area of a sphere with radius \( r \) to its volume, 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 the formula:
\[ A = 4\pi r^2 \]
The volume \( V \) of a sphere is given by the formula:
\[ V = \frac{4}{3}\pi r^3 \]

### Step 2: Set up the ratio of surface area to volume.
We need to find the ratio of the surface area to the volume:
\[ \text{Ratio} = \frac{A}{V} = \frac{4\pi r^2}{\frac{4}{3}\pi r^3} \]

### Step 3: Simplify the ratio.
First, we can cancel out \( \pi \) from the numerator and the denominator:
\[ \text{Ratio} = \frac{4r^2}{\frac{4}{3}r^3} \]
Next, we can simplify the fraction by multiplying by the reciprocal of the denominator:
\[ \text{Ratio} = 4r^2 \times \frac{3}{4r^3} \]
Now, the \( 4 \) in the numerator and denominator cancels out:
\[ \text{Ratio} = 3 \times \frac{r^2}{r^3} \]

### Step 4: Further simplify the expression.
We can simplify \( \frac{r^2}{r^3} \):
\[ \frac{r^2}{r^3} = \frac{1}{r} \]
Thus, we have:
\[ \text{Ratio} = \frac{3}{r} \]

### Final Answer:
The ratio of the surface area of the sphere to its volume is:
\[ \frac{3}{r} \]