Find the area of the iron sheet required to prepare a cone with base is 24 cm height with base raadius 7 cm .

Options

A. `625 cm^(2)`
B. `690cm^(2)`
C. `704cm^(2)`
D. `729cm^(2)`

Answer

To find the area of the iron sheet required to prepare a cone, we need to calculate the total surface area of the cone. The formula for the total surface area (TSA) of a cone is given by:

\[ \text{TSA} = \pi r (r + l) \]

where:
- \( r \) is the radius of the base of the cone,
- \( l \) is the slant height of the cone,
- \( \pi \) is approximately \( \frac{22}{7} \).

### Step 1: Identify the given values
From the problem, we have:
- Radius \( r = 7 \, \text{cm} \)
- Height \( h = 24 \, \text{cm} \)

### Step 2: Calculate the slant height \( l \)
The slant height can be calculated using the Pythagorean theorem:

\[ l = \sqrt{r^2 + h^2} \]

Substituting the known values:

\[ l = \sqrt{7^2 + 24^2} = \sqrt{49 + 576} = \sqrt{625} = 25 \, \text{cm} \]

### Step 3: Substitute the values into the TSA formula
Now we can substitute \( r \) and \( l \) into the TSA formula:

\[ \text{TSA} = \pi r (r + l) = \frac{22}{7} \times 7 \times (7 + 25) \]

### Step 4: Simplify the expression
First, calculate \( (7 + 25) \):

\[ 7 + 25 = 32 \]

Now substitute back into the TSA formula:

\[ \text{TSA} = \frac{22}{7} \times 7 \times 32 \]

### Step 5: Cancel out the \( 7 \)
The \( 7 \) in the numerator and denominator cancels out:

\[ \text{TSA} = 22 \times 32 \]

### Step 6: Calculate the final area
Now calculate \( 22 \times 32 \):

\[ 22 \times 32 = 704 \, \text{cm}^2 \]

### Final Answer
The area of the iron sheet required to prepare the cone is:

\[ \text{TSA} = 704 \, \text{cm}^2 \]

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