A conical paper cup 20 cm across the top and 15 cm deep is full of water. The cup springs a leak at the bottom and losses water at 5 cu. cm per minute.
The amount of water (in `cm^(3))` when the hight of water is 3 cm is

Options

A. `4pi`
B. `3pi`
C. `27pi`
D. `2pi`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the dimensions of the cone
We are given a conical cup with:
- Diameter at the top = 20 cm, hence the radius \( R = \frac{20}{2} = 10 \) cm.
- Height of the cone \( H = 15 \) cm.

### Step 2: Establish a relationship between the radius and height of the water
Let the height of the water be \( h \) cm and the corresponding radius at that height be \( r \) cm. From the properties of similar triangles, we have:

\[ \frac{r}{h} = \frac{R}{H} \]

Substituting the known values:

\[ \frac{r}{h} = \frac{10}{15} = \frac{2}{3} \]

Thus, we can express the radius \( r \) in terms of height \( h \):

\[ r = \frac{2}{3}h \]

### Step 3: Calculate the volume of water when the height is 3 cm
When the height of the water \( h = 3 \) cm, we can find the radius \( r \):

\[ r = \frac{2}{3} \times 3 = 2 \text{ cm} \]

Now, we can calculate the volume \( V \) of the water in the cone using the formula for the volume of a cone:

\[ V = \frac{1}{3} \pi r^2 h \]

Substituting \( r = 2 \) cm and \( h = 3 \) cm into the volume formula:

\[ V = \frac{1}{3} \pi (2)^2 (3) \]

Calculating this gives:

\[ V = \frac{1}{3} \pi (4) (3) = \frac{12}{3} \pi = 4\pi \text{ cm}^3 \]

### Final Answer
The amount of water when the height of water is 3 cm is \( 4\pi \) cm³.

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