The radius of a semicircle is equal to the radius of a sphere whose surface area is 616 cm2 and height of a cylinder is 150% more than radius of semicircle and ratio of height to radius of cylinder is 5: 1. Then find radius of cylinder? (in cm)

Options

A. 14
B. 7.5
C. 3
D. 3.5

Answer

To find the radius of the cylinder, we will follow these steps:

### Step 1: Find the radius of the sphere
The surface area \( A \) of a sphere is given by the formula:
\[ A = 4\pi r^2 \]
We know the surface area of the sphere is \( 616 \, \text{cm}^2 \). Therefore, we can set up the equation:
\[ 4\pi r^2 = 616 \]
To find \( r^2 \), we divide both sides by \( 4\pi \):
\[ r^2 = \frac{616}{4\pi} \]

### Step 2: Substitute the value of \( \pi \)
Using \( \pi \approx \frac{22}{7} \):
\[ r^2 = \frac{616}{4 \times \frac{22}{7}} = \frac{616 \times 7}{88} \]
Calculating this gives:
\[ r^2 = \frac{4312}{88} = 49 \]

### Step 3: Find the radius \( r \)
Taking the square root of both sides:
\[ r = \sqrt{49} = 7 \, \text{cm} \]
Thus, the radius of the semicircle is also \( 7 \, \text{cm} \).

### Step 4: Calculate the height of the cylinder
The height \( h \) of the cylinder is given to be \( 150\% \) more than the radius of the semicircle. This means:
\[ h = r + 1.5r = 2.5r \]
Substituting the value of \( r \):
\[ h = 2.5 \times 7 = 17.5 \, \text{cm} \]

### Step 5: Use the ratio of height to radius of the cylinder
We are given that the ratio of height to radius of the cylinder is \( 5:1 \). Let the radius of the cylinder be \( R \). Then:
\[ \frac{h}{R} = 5 \implies h = 5R \]
Substituting the value of \( h \):
\[ 17.5 = 5R \]

### Step 6: Solve for \( R \)
Dividing both sides by \( 5 \):
\[ R = \frac{17.5}{5} = 3.5 \, \text{cm} \]

Thus, the radius of the cylinder is \( 3.5 \, \text{cm} \).

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