Curved surface area of a cone whose radius is 42 cm is 7656 cm2. If height of cone is half of perimeter of a square, then find the area of square?

Options

A. A)400 `cm^(2)`
B. B)441 `cm^(2)`
C. C)1600 `cm^(2)`
D. D)900 `cm^(2)`

Answer

To solve the problem step by step, we will follow the given information and apply the relevant formulas.

### Step 1: Understand the Curved Surface Area of the Cone
The formula for the curved surface area (CSA) of a cone is given by:
\[ \text{CSA} = \pi r l \]
where \( r \) is the radius and \( l \) is the slant height.

### Step 2: Substitute the Known Values
We know the radius \( r = 42 \) cm and the CSA is \( 7656 \) cm². We will use the value of \( \pi \) as \( \frac{22}{7} \):
\[ 7656 = \frac{22}{7} \times 42 \times l \]

### Step 3: Solve for Slant Height \( l \)
First, we simplify the equation:
\[ 7656 = \frac{22 \times 42}{7} \times l \]
Calculating \( \frac{22 \times 42}{7} \):
\[ \frac{22 \times 42}{7} = \frac{924}{7} = 132 \]
Now, substituting back into the equation:
\[ 7656 = 132l \]
To find \( l \), divide both sides by 132:
\[ l = \frac{7656}{132} = 58 \text{ cm} \]

### Step 4: Find the Height of the Cone
We know the relationship between the height \( h \), radius \( r \), and slant height \( l \) of the cone:
\[ l^2 = r^2 + h^2 \]
Substituting the known values:
\[ 58^2 = 42^2 + h^2 \]
Calculating \( 58^2 \) and \( 42^2 \):
\[ 3364 = 1764 + h^2 \]
Now, isolate \( h^2 \):
\[ h^2 = 3364 - 1764 = 1600 \]
Taking the square root gives:
\[ h = \sqrt{1600} = 40 \text{ cm} \]

### Step 5: Relate Height to the Perimeter of the Square
According to the problem, the height of the cone is half the perimeter of a square:
\[ h = \frac{1}{2} \times \text{Perimeter of Square} \]
The perimeter \( P \) of a square with side \( s \) is given by:
\[ P = 4s \]
Thus, we have:
\[ 40 = \frac{1}{2} \times 4s \]
This simplifies to:
\[ 40 = 2s \implies s = \frac{40}{2} = 20 \text{ cm} \]

### Step 6: Calculate the Area of the Square
The area \( A \) of the square is given by:
\[ A = s^2 = 20^2 = 400 \text{ cm}^2 \]

### Final Answer
The area of the square is \( 400 \text{ cm}^2 \).

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