If the ratio of the diameters of two spheres is 3:5, then what is the ratio of their surface areas?

Options

A. `9:25`
B. `9:10`
C. `3:5`
D. `27:125`

Answer

To solve the problem of finding the ratio of the surface areas of two spheres given the ratio of their diameters, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Given Ratio of Diameters**:
We are given that the ratio of the diameters of two spheres is 3:5. Let's denote:
- Diameter of Sphere 1 (d1) = 3x
- Diameter of Sphere 2 (d2) = 5x

2. **Calculate the Radii of the Spheres**:
The radius of a sphere is half of its diameter. Therefore:
- Radius of Sphere 1 (R1) = d1/2 = (3x)/2 = 1.5x
- Radius of Sphere 2 (R2) = d2/2 = (5x)/2 = 2.5x

3. **Write the Formula for Surface Area of a Sphere**:
The surface area (SA) of a sphere is given by the formula:
\[ SA = 4\pi r^2 \]
where r is the radius of the sphere.

4. **Calculate the Surface Areas of Both Spheres**:
- Surface Area of Sphere 1 (SA1) = \( 4\pi (R1)^2 = 4\pi (1.5x)^2 = 4\pi (2.25x^2) = 9\pi x^2 \)
- Surface Area of Sphere 2 (SA2) = \( 4\pi (R2)^2 = 4\pi (2.5x)^2 = 4\pi (6.25x^2) = 25\pi x^2 \)

5. **Find the Ratio of the Surface Areas**:
Now we can find the ratio of the surface areas of the two spheres:
\[ \text{Ratio of Surface Areas} = \frac{SA1}{SA2} = \frac{9\pi x^2}{25\pi x^2} \]
The \(\pi x^2\) terms cancel out:
\[ \text{Ratio of Surface Areas} = \frac{9}{25} \]

6. **Final Answer**:
Therefore, the ratio of the surface areas of the two spheres is \(9:25\).