A hemispherical cup of radius 4 cm is filled to the brim with coffee. The coffee is then poured into a vertical cone of radius 8 cm and height 16 cm. The percentage of the volume of the cone that remains empty is :

Options

A. 87.5 %`
B. `80.5%`
C. `81.6%`
D. `88.2%`

Answer

To solve the problem step by step, we need to calculate the volumes of the hemispherical cup and the cone, and then determine the percentage of the volume of the cone that remains empty after pouring the coffee from the cup into the cone.

### Step 1: Calculate the volume of the hemispherical cup.

The formula for the volume \( V \) of a hemisphere is given by:

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

Where \( r \) is the radius of the hemisphere. In this case, the radius \( r = 4 \) cm.

Substituting the value of \( r \):

\[ V = \frac{2}{3} \pi (4)^3 = \frac{2}{3} \pi (64) = \frac{128}{3} \pi \text{ cm}^3 \]

### Step 2: Calculate the volume of the cone.

The formula for the volume \( V \) of a cone is given by:

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

Where \( r \) is the radius and \( h \) is the height of the cone. Here, the radius \( r = 8 \) cm and the height \( h = 16 \) cm.

Substituting the values:

\[ V = \frac{1}{3} \pi (8)^2 (16) = \frac{1}{3} \pi (64)(16) = \frac{1}{3} \pi (1024) = \frac{1024}{3} \pi \text{ cm}^3 \]

### Step 3: Determine the volume of the cone that remains empty after pouring the coffee.

The volume of coffee poured into the cone is equal to the volume of the hemispherical cup:

\[ \text{Volume of coffee} = \frac{128}{3} \pi \text{ cm}^3 \]

Now, we need to find the volume of the cone that remains empty. We do this by subtracting the volume of coffee from the total volume of the cone:

\[ \text{Volume remaining in cone} = \text{Volume of cone} - \text{Volume of coffee} \]

Substituting the values:

\[ \text{Volume remaining in cone} = \frac{1024}{3} \pi - \frac{128}{3} \pi = \left(\frac{1024 - 128}{3}\right) \pi = \frac{896}{3} \pi \text{ cm}^3 \]

### Step 4: Calculate the percentage of the volume of the cone that remains empty.

To find the percentage of the volume of the cone that remains empty, we use the formula:

\[ \text{Percentage empty} = \left(\frac{\text{Volume remaining}}{\text{Total volume of cone}}\right) \times 100 \]

Substituting the values:

\[ \text{Percentage empty} = \left(\frac{\frac{896}{3} \pi}{\frac{1024}{3} \pi}\right) \times 100 \]

The \( \pi \) and \( \frac{1}{3} \) cancel out:

\[ \text{Percentage empty} = \left(\frac{896}{1024}\right) \times 100 \]

Calculating this gives:

\[ \text{Percentage empty} = 87.5\% \]

### Final Answer:

The percentage of the volume of the cone that remains empty is **87.5%**.

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