The total surface area of a right circular cone of slant height 13 cm is `90pi cm^(2).` Calculate its volume in `cm^(3).` (Take `pi = 3.14`).

Answer

To solve the problem step by step, we will follow the mathematical process outlined in the video transcript.

### Step 1: Understand the Given Information
We are given:
- Slant height (L) of the cone = 13 cm
- Total surface area (TSA) of the cone = 90π cm²

### Step 2: Use the Formula for Total Surface Area of a Cone
The formula for the total surface area (TSA) of a right circular cone is:
\[ \text{TSA} = \pi r (l + r) \]
where \( r \) is the radius and \( l \) is the slant height.

### Step 3: Substitute the Known Values into the Formula
Substituting the known values into the TSA formula:
\[ 90\pi = \pi r (13 + r) \]

### Step 4: Simplify the Equation
We can cancel π from both sides:
\[ 90 = r (13 + r) \]
This simplifies to:
\[ 90 = 13r + r^2 \]

### Step 5: Rearrange the Equation
Rearranging the equation gives us:
\[ r^2 + 13r - 90 = 0 \]

### Step 6: Factor the Quadratic Equation
To factor the quadratic equation, we look for two numbers that multiply to -90 and add to 13. The numbers are 18 and -5:
\[ r^2 + 18r - 5r - 90 = 0 \]
Factoring by grouping:
\[ (r + 18)(r - 5) = 0 \]

### Step 7: Solve for r
Setting each factor to zero gives us:
1. \( r + 18 = 0 \) → \( r = -18 \) (not possible since radius cannot be negative)
2. \( r - 5 = 0 \) → \( r = 5 \) cm

### Step 8: Find the Height of the Cone
Now we need to find the height (h) of the cone using the formula:
\[ h = \sqrt{l^2 - r^2} \]
Substituting the values:
\[ h = \sqrt{13^2 - 5^2} = \sqrt{169 - 25} = \sqrt{144} = 12 \text{ cm} \]

### Step 9: Calculate the Volume of the Cone
The volume (V) of the cone is given by the formula:
\[ V = \frac{1}{3} \pi r^2 h \]
Substituting the values:
\[ V = \frac{1}{3} \times 3.14 \times 5^2 \times 12 \]
Calculating step by step:
\[ V = \frac{1}{3} \times 3.14 \times 25 \times 12 \]
\[ = \frac{1}{3} \times 3.14 \times 300 \]
\[ = \frac{942}{3} = 314 \text{ cm}^3 \]

### Final Answer
Thus, the volume of the cone is:
\[ \text{Volume} = 314 \text{ cm}^3 \]

---