The volume of a solid right circular cone is `600picm^3`, and the diameter of its base is 30 cm. The total surface area (in `cm^2` ) of the cone is:

Options

A. `496pi`
B. `480pi`
C. `255pi`
D. `472pi`

Answer

To solve the problem step by step, we will find the total surface area of the cone using the given volume and diameter.

### Step 1: Identify the given values
- Volume of the cone, \( V = 600 \pi \, \text{cm}^3 \)
- Diameter of the base, \( d = 30 \, \text{cm} \)

### Step 2: Calculate the radius of the base
The radius \( r \) is half of the diameter:
\[ r = \frac{d}{2} = \frac{30}{2} = 15 \, \text{cm} \]

### Step 3: Use the volume formula to find the height
The formula for the volume of a cone is:
\[ V = \frac{1}{3} \pi r^2 h \]
Substituting the known values:
\[ 600 \pi = \frac{1}{3} \pi (15^2) h \]
This simplifies to:
\[ 600 = \frac{1}{3} (15^2) h \]
Calculating \( 15^2 \):
\[ 15^2 = 225 \]
So we have:
\[ 600 = \frac{1}{3} \times 225 \times h \]
Multiplying both sides by 3:
\[ 1800 = 225h \]
Now, divide by 225 to find \( h \):
\[ h = \frac{1800}{225} = 8 \, \text{cm} \]

### Step 4: Calculate the slant height \( l \)
The slant height \( l \) can be found using the Pythagorean theorem:
\[ l = \sqrt{r^2 + h^2} \]
Substituting the values of \( r \) and \( h \):
\[ l = \sqrt{15^2 + 8^2} = \sqrt{225 + 64} = \sqrt{289} = 17 \, \text{cm} \]

### Step 5: Calculate the total surface area
The total surface area \( A \) of a cone is given by:
\[ A = \pi r l + \pi r^2 \]
Factoring out \( \pi r \):
\[ A = \pi r (l + r) \]
Substituting the known values:
\[ A = \pi \times 15 \times (17 + 15) = \pi \times 15 \times 32 \]
Calculating:
\[ A = 480 \pi \, \text{cm}^2 \]

### Final Answer
The total surface area of the cone is \( 480 \pi \, \text{cm}^2 \).

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