Find the total surface area of a right circular cone whose slant height is 25 cm and area of base is `154 cm^(2)`.

Answer

To find the total surface area of a right circular cone, we will follow these steps:

### Step 1: Understand the formula for total surface area of a cone
The total surface area (TSA) of a right circular cone is given by the formula:
\[ \text{TSA} = \pi r^2 + \pi r l \]
where:
- \( r \) is the radius of the base of the cone,
- \( l \) is the slant height of the cone.

### Step 2: Identify the given values
From the problem, we know:
- Slant height \( l = 25 \, \text{cm} \)
- Area of the base \( = 154 \, \text{cm}^2 \)

### Step 3: Relate the area of the base to the radius
The area of the base of the cone is given by:
\[ \text{Area of base} = \pi r^2 \]
Since the area of the base is given as \( 154 \, \text{cm}^2 \), we can set up the equation:
\[ \pi r^2 = 154 \]
Using \( \pi \approx \frac{22}{7} \), we can write:
\[ \frac{22}{7} r^2 = 154 \]

### Step 4: Solve for \( r^2 \)
To find \( r^2 \), we multiply both sides by \( \frac{7}{22} \):
\[ r^2 = 154 \times \frac{7}{22} \]
Calculating this gives:
\[ r^2 = 154 \times \frac{7}{22} = 49 \]

### Step 5: Calculate \( r \)
Now, take the square root of \( r^2 \) to find \( r \):
\[ r = \sqrt{49} = 7 \, \text{cm} \]

### Step 6: Substitute \( r \) and \( l \) into the TSA formula
Now that we have \( r \) and \( l \), we can substitute these values into the TSA formula:
\[ \text{TSA} = \pi r^2 + \pi r l \]
Substituting the values:
\[ \text{TSA} = \frac{22}{7} \times 7^2 + \frac{22}{7} \times 7 \times 25 \]

### Step 7: Calculate each term
Calculating the first term:
\[ \frac{22}{7} \times 7^2 = \frac{22}{7} \times 49 = 154 \]
Calculating the second term:
\[ \frac{22}{7} \times 7 \times 25 = \frac{22 \times 25}{1} = 550 \]

### Step 8: Add both terms to find TSA
Now, add both terms together:
\[ \text{TSA} = 154 + 550 = 704 \, \text{cm}^2 \]

### Final Answer
The total surface area of the right circular cone is:
\[ \text{TSA} = 704 \, \text{cm}^2 \]

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