The diameter of two hollow sphere made from the same metal sheet are 21 cm and 17.5 cm respectively. The ratio of the area of metal sheets required for making the two sphere is

Options

A. `6:5`
B. `36:25`
C. `3:2`
D. `18:25`

Answer

To find the ratio of the area of metal sheets required for making the two hollow spheres, we need to calculate the surface area of each sphere and then find the ratio of these areas.

### Step 1: Find the radius of each sphere.
- The diameter of the first sphere is 21 cm, so the radius \( r_1 \) is:
\[ r_1 = \frac{21}{2} = 10.5 \text{ cm} \]
- The diameter of the second sphere is 17.5 cm, so the radius \( r_2 \) is:
\[ r_2 = \frac{17.5}{2} = 8.75 \text{ cm} \]

### Step 2: Calculate the surface area of each sphere.
The formula for the surface area \( A \) of a sphere is given by:
\[ A = 4\pi r^2 \]
- For the first sphere:
\[ A_1 = 4\pi (10.5)^2 = 4\pi \times 110.25 = 441\pi \text{ cm}^2 \]
- For the second sphere:
\[ A_2 = 4\pi (8.75)^2 = 4\pi \times 76.5625 = 306.25\pi \text{ cm}^2 \]

### Step 3: Find the ratio of the surface areas.
To find the ratio of the areas \( A_1 \) and \( A_2 \):
\[ \text{Ratio} = \frac{A_1}{A_2} = \frac{441\pi}{306.25\pi} = \frac{441}{306.25} \]
To simplify this ratio, we can multiply both the numerator and the denominator by 100 to eliminate the decimal:
\[ \frac{44100}{30625} \]
Now, we can simplify this fraction. Dividing both by 25:
\[ \frac{1764}{1225} \]
Now, we can further simplify this ratio:
\[ \frac{1764 \div 1764}{1225 \div 1764} = \frac{1}{0.693} \]
This gives us a ratio of approximately:
\[ \frac{441}{306.25} \approx \frac{1764}{1225} \]

### Final Step: Write the final ratio.
The ratio of the area of metal sheets required for making the two spheres is:
\[ \frac{441}{306.25} \approx \frac{1764}{1225} \]