Ratio of volume of conical tent to total surface area of hemispherical bowl is 8: 1. Radius of bowl and tent is equal and slant height of tent is 75cm. If radius of conical tent is equal to side of a cube, then find curved surface area of cube.

Options

A. A)`1576 cm ^(2)`
B. B)`1764 cm ^(2)`
C. C)`1852 cm ^(2)`
D. D)`1690 cm ^(2)`

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the given information
We know that:
- The ratio of the volume of the conical tent to the total surface area of the hemispherical bowl is 8:1.
- The radius of the conical tent and the hemispherical bowl is equal.
- The slant height of the conical tent is 75 cm.
- The radius of the conical tent is equal to the side of a cube.

### Step 2: Write the formulas
1. **Volume of the conical tent (V)**:
\[ V = \frac{1}{3} \pi r^2 h \]
2. **Total surface area of the hemispherical bowl (A)**:
\[ A = 3 \pi r^2 \]

### Step 3: Set up the ratio
Given the ratio of the volume of the conical tent to the total surface area of the hemispherical bowl:
\[ \frac{V}{A} = \frac{8}{1} \]
Substituting the formulas:
\[ \frac{\frac{1}{3} \pi r^2 h}{3 \pi r^2} = \frac{8}{1} \]

### Step 4: Simplify the equation
Cancel out \(\pi r^2\) from both sides:
\[ \frac{h}{9} = 8 \]
Multiply both sides by 9:
\[ h = 72 \text{ cm} \]

### Step 5: Use the slant height to find the radius
We know the slant height \(l\) of the conical tent is 75 cm. We can use the Pythagorean theorem to find the radius \(r\):
\[ l^2 = r^2 + h^2 \]
Substituting the known values:
\[ 75^2 = r^2 + 72^2 \]
Calculating:
\[ 5625 = r^2 + 5184 \]
\[ r^2 = 5625 - 5184 = 441 \]
\[ r = \sqrt{441} = 21 \text{ cm} \]

### Step 6: Find the side of the cube
Since the radius of the conical tent is equal to the side of the cube:
\[ \text{Side of the cube} = r = 21 \text{ cm} \]

### Step 7: Calculate the curved surface area of the cube
The formula for the curved surface area (CSA) of a cube is:
\[ \text{CSA} = 4a^2 \]
Substituting the side length:
\[ \text{CSA} = 4 \times (21)^2 = 4 \times 441 = 1764 \text{ cm}^2 \]

### Final Answer
The curved surface area of the cube is **1764 cm²**.

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