Two spheres are cut from a certain uniform rock. One has radius 4.50cm. The mass of the other is five times greater. Find its radius. Give your answer to 2 decimal places
First, let's find the mass of the sphere with radius 4.50cm. The formula for the mass of a sphere is:
\[ M = \frac{4}{3} \pi r^3 \rho \]
where M is the mass, r is the radius, and ρ is the density of the material.
Let's assume the density of the material is ρ and the mass of the first sphere is m. So,
\[ m = \frac{4}{3} \pi (4.50)^3 \rho \]
Now, the mass of the second sphere (5 times greater) is 5m. Using the same formula, we get:
\[ 5m = \frac{4}{3} \pi r^3 \rho \]
Dividing the second equation by the first:
\[ 5 = \frac{r^3}{(4.50)^3} \]
\[ r = (5 \times (4.50)^3)^{\frac{1}{3}} \]
\[ r \approx 6.92 \]
Therefore, the radius of the second sphere is approximately 6.92 cm.