The area of the base of a conical solid is `38.5 cm^(2)` and its volume is `154 cm^(3).` Find the curved surface area of the solid.

Answer

To solve the problem step by step, we will use the given data about the conical solid to find its curved surface area.

### Step 1: Identify the area of the base
The area of the base of the cone is given as:
\[ \text{Area of base} = 38.5 \, \text{cm}^2 \]
The base of a cone is circular, so we can use the formula for the area of a circle:
\[ \text{Area} = \pi r^2 \]
Where \( r \) is the radius of the base.

### Step 2: Set up the equation for the radius
Substituting the area into the formula, we have:
\[ \pi r^2 = 38.5 \]
Using \( \pi \approx \frac{22}{7} \), we can rewrite the equation as:
\[ \frac{22}{7} r^2 = 38.5 \]

### Step 3: Solve for \( r^2 \)
To eliminate the fraction, multiply both sides by \( 7 \):
\[ 22 r^2 = 38.5 \times 7 \]
Calculating \( 38.5 \times 7 \):
\[ 38.5 \times 7 = 269.5 \]
So we have:
\[ 22 r^2 = 269.5 \]
Now, divide both sides by \( 22 \):
\[ r^2 = \frac{269.5}{22} = 12.25 \]

### Step 4: Calculate the radius \( r \)
Taking the square root of both sides:
\[ r = \sqrt{12.25} = 3.5 \, \text{cm} \]

### Step 5: Use the volume formula to find the height \( h \)
The volume \( V \) of a cone is given by:
\[ V = \frac{1}{3} \pi r^2 h \]
We know the volume is \( 154 \, \text{cm}^3 \), so we set up the equation:
\[ 154 = \frac{1}{3} \times \frac{22}{7} \times (3.5)^2 \times h \]
Calculating \( (3.5)^2 \):
\[ (3.5)^2 = 12.25 \]
Substituting this back into the volume equation:
\[ 154 = \frac{1}{3} \times \frac{22}{7} \times 12.25 \times h \]
Now, simplify:
\[ 154 = \frac{22 \times 12.25}{21} h \]
Calculating \( \frac{22 \times 12.25}{21} \):
\[ \frac{22 \times 12.25}{21} = \frac{270.5}{21} \approx 12.9 \]
So we have:
\[ 154 = 12.9 h \]
Now, solve for \( h \):
\[ h = \frac{154}{12.9} \approx 11.95 \, \text{cm} \]

### Step 6: Calculate the slant height \( l \)
The slant height \( l \) can be calculated using the Pythagorean theorem:
\[ l = \sqrt{h^2 + r^2} \]
Substituting the values of \( h \) and \( r \):
\[ l = \sqrt{(11.95)^2 + (3.5)^2} \]
Calculating \( (11.95)^2 \) and \( (3.5)^2 \):
\[ (11.95)^2 \approx 142.62, \quad (3.5)^2 = 12.25 \]
So:
\[ l = \sqrt{142.62 + 12.25} = \sqrt{154.87} \approx 12.43 \, \text{cm} \]

### Step 7: Calculate the curved surface area
The curved surface area \( A \) of a cone is given by:
\[ A = \pi r l \]
Substituting the values of \( r \) and \( l \):
\[ A = \frac{22}{7} \times 3.5 \times 12.43 \]
Calculating:
\[ A \approx \frac{22 \times 3.5 \times 12.43}{7} \approx \frac{22 \times 43.505}{7} \approx 132.5 \, \text{cm}^2 \]

### Final Answer
The curved surface area of the conical solid is approximately:
\[ \text{Curved Surface Area} \approx 132.5 \, \text{cm}^2 \]