To find the density of the sphere, we need to use the formula for density, which is:
\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} \]
Given data:
- Mass = 100 g
- Surface area = 39 cm²
First, we need to find the radius of the sphere using the surface area formula for a sphere:
\[ \text{Surface Area} = 4\pi r^2 \]
Setting the surface area to 39 cm²:
\[ 4\pi r^2 = 39 \]
Now, we can solve for \( r^2 \):
\[ r^2 = \frac{39}{4\pi} \]
Calculating \( r^2 \):
\[ r^2 \approx \frac{39}{4 \times 3.14159} \approx \frac{39}{12.56636} \approx 3.104 \] \[ r \approx \sqrt{3.104} \approx 1.76 \text{ cm} \]
Now, we can find the volume of the sphere using the volume formula:
\[ \text{Volume} = \frac{4}{3} \pi r^3 \]
Calculating \( r^3 \):
\[ r^3 \approx (1.76)^3 \approx 5.44 \text{ cm}^3 \]
Now plug \( r^3 \) into the volume formula:
\[ \text{Volume} \approx \frac{4}{3} \pi (5.44) \approx \frac{4}{3} \times 3.14159 \times 5.44 \approx 22.78 \text{ cm}^3 \]
Now we can find the density:
\[ \text{Density} = \frac{\text{Mass}}{\text{Volume}} = \frac{100 \text{ g}}{22.78 \text{ cm}^3} \approx 4.39 \text{ g/cm}^3 \]
Therefore, the density of the sphere is approximately \( 4.39 , \text{g/cm}^3 \).