If the area of the base of a cone is `770 cm^(2)` and the area of the curved surface is `814 cm^(2)`, then its volume (in `cm^(3)`) is :

Options

A. `213 sqrt(5)`
B. `392sqrt(5)`
C. `550 sqrt(5)`
D. `616 sqrt(5)`

Answer

To find the volume of the cone given the area of the base and the area of the curved surface, we can follow these steps:

### Step 1: Calculate the radius of the base of the cone
We know that the area of the base of the cone (which is a circle) is given by the formula:
\[ \text{Area} = \pi r^2 \]
Given that the area of the base is \( 770 \, \text{cm}^2 \):
\[ \pi r^2 = 770 \]
Using \( \pi \approx \frac{22}{7} \):
\[ \frac{22}{7} r^2 = 770 \]
Multiplying both sides by \( \frac{7}{22} \):
\[ r^2 = 770 \times \frac{7}{22} \]
Calculating the right side:
\[ r^2 = 35 \times 7 = 245 \]
Taking the square root:
\[ r = \sqrt{245} = 7\sqrt{5} \, \text{cm} \]

### Step 2: Calculate the slant height of the cone
The area of the curved surface of the cone is given by the formula:
\[ \text{Curved Surface Area} = \pi r l \]
Given that the curved surface area is \( 814 \, \text{cm}^2 \):
\[ \pi r l = 814 \]
Substituting \( r = 7\sqrt{5} \):
\[ \frac{22}{7} \times 7\sqrt{5} \times l = 814 \]
This simplifies to:
\[ 22\sqrt{5} l = 814 \]
Dividing both sides by \( 22\sqrt{5} \):
\[ l = \frac{814}{22\sqrt{5}} \]
Calculating \( l \):
\[ l = \frac{37}{\sqrt{5}} \, \text{cm} \]

### Step 3: Calculate the height of the cone
Using the Pythagorean theorem in the cone:
\[ l^2 = h^2 + r^2 \]
Substituting the values:
\[ \left(\frac{37}{\sqrt{5}}\right)^2 = h^2 + (7\sqrt{5})^2 \]
Calculating the squares:
\[ \frac{1369}{5} = h^2 + 245 \]
Rearranging gives:
\[ h^2 = \frac{1369}{5} - 245 \]
Converting \( 245 \) to a fraction:
\[ 245 = \frac{1225}{5} \]
Thus:
\[ h^2 = \frac{1369 - 1225}{5} = \frac{144}{5} \]
Taking the square root:
\[ h = \frac{12}{\sqrt{5}} \, \text{cm} \]

### Step 4: Calculate the volume of the cone
The volume \( V \) of the cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
Substituting the values:
\[ V = \frac{1}{3} \times \frac{22}{7} \times (7\sqrt{5})^2 \times \frac{12}{\sqrt{5}} \]
Calculating \( (7\sqrt{5})^2 = 245 \):
\[ V = \frac{1}{3} \times \frac{22}{7} \times 245 \times \frac{12}{\sqrt{5}} \]
Simplifying:
\[ V = \frac{1}{3} \times 22 \times 35 \times \frac{12}{\sqrt{5}} \]
Calculating:
\[ V = \frac{1}{3} \times 770 \times \frac{12}{\sqrt{5}} \]
\[ V = \frac{9240}{3\sqrt{5}} = \frac{3080}{\sqrt{5}} \]
To rationalize:
\[ V = 3080 \times \frac{\sqrt{5}}{5} = \frac{3080\sqrt{5}}{5} = 616\sqrt{5} \, \text{cm}^3 \]

Thus, the volume of the cone is \( 616\sqrt{5} \, \text{cm}^3 \).