If the area of the base of a cylinder is `346.5cm^(2)` and the area of the curved surface is `990cm^(2)`, then its height is: (Take `pi= (22)/(7)`)

Options

A. 10cm
B. 12cm
C. 14cm
D. 15cm

Answer

To find the height of the cylinder, we can follow these steps:

### Step 1: Understand the formulas
The area of the base of a cylinder is given by the formula:
\[ \text{Area of base} = \pi r^2 \]
The curved surface area (CSA) of a cylinder is given by:
\[ \text{CSA} = 2 \pi r h \]

### Step 2: Use the area of the base to find the radius
We know the area of the base is \(346.5 \, \text{cm}^2\). Using the formula for the area of the base:
\[ \pi r^2 = 346.5 \]
Substituting \(\pi = \frac{22}{7}\):
\[ \frac{22}{7} r^2 = 346.5 \]
To eliminate the fraction, multiply both sides by \(7\):
\[ 22 r^2 = 346.5 \times 7 \]
Calculating the right side:
\[ 346.5 \times 7 = 2425.5 \]
So we have:
\[ 22 r^2 = 2425.5 \]
Now, divide both sides by \(22\):
\[ r^2 = \frac{2425.5}{22} \]
Calculating the division:
\[ r^2 = 110.25 \]
Taking the square root to find \(r\):
\[ r = \sqrt{110.25} = 10.5 \, \text{cm} \]

### Step 3: Use the curved surface area to find the height
Now we can use the curved surface area to find the height \(h\):
\[ 2 \pi r h = 990 \]
Substituting \(\pi = \frac{22}{7}\) and \(r = 10.5\):
\[ 2 \times \frac{22}{7} \times 10.5 \times h = 990 \]
Calculating \(2 \times \frac{22}{7} \times 10.5\):
\[ = \frac{44 \times 10.5}{7} = \frac{462}{7} = 66 \]
So we have:
\[ 66h = 990 \]
Now, divide both sides by \(66\):
\[ h = \frac{990}{66} \]
Calculating the division:
\[ h = 15 \, \text{cm} \]

### Final Answer
The height of the cylinder is \(15 \, \text{cm}\).

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