A water fountain on the ground sprinkles water all around it. If the speed of water coming out of the fountains is v, the total area around the fountain that gets wet is:

Options

A. ` pi (v ^ 2 ) /( g ) `
B. ` pi (v^4 ) /(g ^ 2 ) `
C. ` (pi ) /(2) (v^4 )/(g ^ 4 ) `
D. ` pi (v^ 2 ) /(g ^ 2 ) `

Explanation

To find the total area around a water fountain that gets wet, we can follow these steps:

### Step 1: Understand the Problem
The problem states that a water fountain sprinkles water in all directions with a speed \( v \). We need to determine the total area that gets wet.

**Hint:** Visualize the fountain as a source of water that spreads out in a circular pattern.

### Step 2: Determine the Range of the Water
The water from the fountain behaves like a projectile. To maximize the horizontal distance (range) that the water can reach, it should be ejected at an angle of 45 degrees. The formula for the range \( R \) of a projectile is given by:

\[ R = \frac{u^2 \sin(2\theta)}{g} \]

where:
- \( u \) is the initial speed (in this case, \( v \)),
- \( \theta \) is the angle of projection (45 degrees),
- \( g \) is the acceleration due to gravity.

**Hint:** Remember that the sine of 90 degrees (which is \( 2 \times 45 \)) is 1.

### Step 3: Calculate the Maximum Range
Substituting \( u = v \) and \( \theta = 45^\circ \) into the range formula, we get:

\[ R_{\text{max}} = \frac{v^2 \cdot 1}{g} = \frac{v^2}{g} \]

**Hint:** This tells you how far the water can reach horizontally from the fountain.

### Step 4: Determine the Area that Gets Wet
The area that gets wet can be modeled as a circle with radius equal to the maximum range \( R_{\text{max}} \). The area \( A \) of a circle is given by the formula:

\[ A = \pi R^2 \]

Substituting \( R_{\text{max}} \) into the area formula:

\[ A = \pi \left( \frac{v^2}{g} \right)^2 \]

### Step 5: Simplify the Area Expression
Now, simplify the expression for the area:

\[ A = \pi \frac{v^4}{g^2} \]

**Hint:** This is the final expression for the area that gets wet.

### Final Answer
The total area around the fountain that gets wet is:

\[ A = \frac{\pi v^4}{g^2} \]

This matches with option number 2 if given in a multiple-choice format.