The sum of the radius and height of a right circular cylinder (A), is 31 metres. The total surface area of the cylinder is 1364 `M^2`. If the radius of another right circular cylinder (B) is 3.5 metres more than the radius of cylinder (A), what is the circumference of base of cylinder (B) ? (in metre)

Options

A. 64
B. 66
C. 80
D. 88

Answer

To solve the problem step by step, we will follow these steps:

### Step 1: Set up the equations
Let the radius of cylinder A be \( r \) meters and the height be \( h \) meters. According to the problem, we have the following equations:
1. \( r + h = 31 \) (The sum of the radius and height of cylinder A)
2. The formula for the total surface area (TSA) of a cylinder is given by:
\[ TSA = 2\pi r(h + r) \]
We know the TSA is 1364 m², so:
\[ 2\pi r(h + r) = 1364 \]

### Step 2: Substitute \( h \) in terms of \( r \)
From the first equation, we can express \( h \) in terms of \( r \):
\[ h = 31 - r \]

### Step 3: Substitute \( h \) into the TSA equation
Now substitute \( h \) into the TSA equation:
\[ 2\pi r((31 - r) + r) = 1364 \]
This simplifies to:
\[ 2\pi r(31) = 1364 \]

### Step 4: Solve for \( r \)
Now, substituting \( \pi \) with \( \frac{22}{7} \):
\[ 2 \cdot \frac{22}{7} \cdot r \cdot 31 = 1364 \]
Multiply both sides by 7 to eliminate the fraction:
\[ 2 \cdot 22 \cdot r \cdot 31 = 1364 \cdot 7 \]
Calculating \( 1364 \cdot 7 \):
\[ 1364 \cdot 7 = 9558 \]
So we have:
\[ 44 \cdot r \cdot 31 = 9558 \]
Now divide both sides by \( 44 \cdot 31 \):
\[ r = \frac{9558}{44 \cdot 31} \]
Calculating \( 44 \cdot 31 = 1364 \):
\[ r = \frac{9558}{1364} = 7 \text{ meters} \]

### Step 5: Find the height \( h \)
Now substitute \( r \) back to find \( h \):
\[ h = 31 - r = 31 - 7 = 24 \text{ meters} \]

### Step 6: Find the radius of cylinder B
The radius of cylinder B is 3.5 meters more than the radius of cylinder A:
\[ r_B = r + 3.5 = 7 + 3.5 = 10.5 \text{ meters} \]

### Step 7: Calculate the circumference of cylinder B
The circumference \( C \) of the base of cylinder B is given by:
\[ C = 2\pi r_B \]
Substituting \( r_B \) and \( \pi \):
\[ C = 2 \cdot \frac{22}{7} \cdot 10.5 \]
Calculating:
\[ C = \frac{44}{7} \cdot 10.5 = \frac{44 \cdot 10.5}{7} = \frac{462}{7} = 66 \text{ meters} \]

### Final Answer
The circumference of the base of cylinder B is **66 meters**.

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