The radius of a solid right circular cylinder is `66 2/3` % of its height . If height is h centimetres then its total surface area `( "in cm"^(2))` is :

Options

A. ` 44/9 pih^(2)`
B. `20/9 pih^(2)`
C. `40/9 pih^(2)`
D. `4pih^(2)`

Answer

To solve the problem, we need to find the total surface area of a solid right circular cylinder given that the radius is \(66 \frac{2}{3}\%\) of its height \(h\).

### Step-by-Step Solution:

1. **Convert the percentage to a fraction**:
\[ 66 \frac{2}{3}\% = \frac{200}{3}\% \]
To convert this percentage to a fraction, we divide by 100:
\[ \frac{200}{3} \times \frac{1}{100} = \frac{200}{300} = \frac{2}{3} \]

2. **Express the radius in terms of height**:
Let the height of the cylinder be \(h\) cm. Then, the radius \(r\) can be expressed as:
\[ r = \frac{2}{3}h \]

3. **Calculate the total surface area of the cylinder**:
The total surface area \(A\) of a right circular cylinder is given by the formula:
\[ A = 2\pi r(h + r) \]
Substituting \(r = \frac{2}{3}h\) into the formula:
\[ A = 2\pi \left(\frac{2}{3}h\right) \left(h + \frac{2}{3}h\right) \]

4. **Simplify the expression**:
First, simplify \(h + \frac{2}{3}h\):
\[ h + \frac{2}{3}h = \frac{3}{3}h + \frac{2}{3}h = \frac{5}{3}h \]
Now substitute this back into the area formula:
\[ A = 2\pi \left(\frac{2}{3}h\right) \left(\frac{5}{3}h\right) \]

5. **Calculate the area**:
\[ A = 2\pi \cdot \frac{2 \cdot 5}{3 \cdot 3} h^2 = 2\pi \cdot \frac{10}{9} h^2 = \frac{20\pi}{9} h^2 \]

Thus, the total surface area of the cylinder is:
\[ \boxed{\frac{20\pi}{9} h^2} \text{ cm}^2 \]