A right circular solid cylinder has radius of base 7 cm and height is 28 cm. It is melted to form a cuboid such that the ratio of its side is 2 : 3 : 6. What is the total surface area `("in" cm^(2))` cuboid ?

Options

A. 72x`(1078)^(2/3)`
B. 36x`(2078)^(2/3)`
C. 72x`(2078)^(2/3)`
D. 36x`(1078)^(2/3)`

Answer

To solve the problem step by step, we need to find the total surface area of the cuboid formed by melting a right circular solid cylinder.

### Step 1: Calculate the Volume of the Cylinder
The volume \( V \) of a right circular cylinder is given by the formula:
\[ V = \pi r^2 h \]
Where:
- \( r \) is the radius of the base
- \( h \) is the height

Given:
- Radius \( r = 7 \) cm
- Height \( h = 28 \) cm

Substituting the values:
\[ V = \pi \times (7)^2 \times 28 = \pi \times 49 \times 28 \]
Using \( \pi \approx \frac{22}{7} \):
\[ V = \frac{22}{7} \times 49 \times 28 \]

### Step 2: Simplify the Volume Calculation
Calculating \( 49 \times 28 \):
\[ 49 \times 28 = 1372 \]
Now substituting back:
\[ V = \frac{22}{7} \times 1372 \]
This simplifies to:
\[ V = 22 \times 196 = 4312 \text{ cm}^3 \]

### Step 3: Set Up the Volume of the Cuboid
Let the dimensions of the cuboid be \( 2x, 3x, \) and \( 6x \). The volume \( V \) of the cuboid is given by:
\[ V = l \times b \times h = (2x) \times (3x) \times (6x) = 36x^3 \]

### Step 4: Equate the Volumes
Since the volume of the cylinder equals the volume of the cuboid:
\[ 4312 = 36x^3 \]
Now, solve for \( x^3 \):
\[ x^3 = \frac{4312}{36} \]

### Step 5: Calculate \( x^3 \)
Calculating \( \frac{4312}{36} \):
\[ x^3 = 119.7777 \text{ (approximately)} \]

### Step 6: Find \( x \)
Taking the cube root:
\[ x = \sqrt[3]{119.7777} \approx 4.93 \text{ cm} \]

### Step 7: Calculate the Total Surface Area of the Cuboid
The total surface area \( TSA \) of a cuboid is given by:
\[ TSA = 2(lb + bh + hl) \]
Substituting \( l = 2x, b = 3x, h = 6x \):
\[ TSA = 2((2x)(3x) + (3x)(6x) + (6x)(2x)) \]
Calculating each term:
- \( (2x)(3x) = 6x^2 \)
- \( (3x)(6x) = 18x^2 \)
- \( (6x)(2x) = 12x^2 \)

So,
\[ TSA = 2(6x^2 + 18x^2 + 12x^2) = 2(36x^2) = 72x^2 \]

### Step 8: Substitute \( x \) into the TSA Formula
Substituting \( x \approx 4.93 \):
\[ TSA = 72(4.93)^2 \]
Calculating \( (4.93)^2 \):
\[ (4.93)^2 \approx 24.3 \]
Thus,
\[ TSA \approx 72 \times 24.3 \approx 1749.6 \text{ cm}^2 \]

### Final Answer
The total surface area of the cuboid is approximately \( 1749.6 \text{ cm}^2 \).