Two spheres are cut from a certain uniform rock.One has radius 4.50 cm.The mass of the other is five times greater. Find its radius.
the Answer is not 4/3pi times the cubed root of 5 times the r^3...it is just cubed root 5 times the radius of 4.5
Since the rock is of uniform density, the ratio of the masses is the same as the ratio of the volumes.
V = 4/3 pi r^3
So, 5 * 4/3 pi r^3 = 4/3 pi * 5 r^3 = 4/3 pi * (cbrt(5)*r)^3
So, the new radius is cube root 5 times the old one.
We can use the formula for the volume of a sphere to solve this problem. The formula for the volume of a sphere is given by:
V = (4/3) * π * r^3
Let's first find the volume of the sphere with the radius of 4.50 cm:
V1 = (4/3) * π * (4.50 cm)^3
Now, let's find the volume of the other sphere, which has five times the mass of the first one:
V2 = 5 * V1
Since the volumes of the two spheres are directly proportional to the cubes of their respective radii, we can set up the following equation:
V2 = (4/3) * π * r2^3
Substituting 5 * V1 for V2, we have:
5 * V1 = (4/3) * π * r2^3
Now, solve for r2:
r2^3 = (5 * V1) / ((4/3) * π)
r2^3 = (5/3) * V1 / π
r2 = ( (5/3) * V1 / π )^(1/3)
Plugging the value of V1 into the equation, we have:
r2 = ( (5/3) * [ (4/3) * π * (4.50 cm)^3 ] / π )^(1/3)
Simplifying further:
r2 = ( (5/3) * (4/3) * (4.50 cm)^3 )^(1/3)
Calculating the right side of the equation:
r2 = ( (5/3) * (4/3) * (4.50 cm)^3 )^(1/3)
r2 ≈ 8.65 cm
Therefore, the radius of the second sphere is approximately 8.65 cm.
To solve this problem, we need to use the concept of mass and volume relationship for spheres.
The mass of a sphere is directly proportional to its volume, and the volume of a sphere is given by the formula:
Volume = (4/3) * π * radius^3
Let's first calculate the volume of the sphere with a radius of 4.50 cm:
Volume1 = (4/3) * π * (4.50 cm)^3
Next, we can set up a ratio of the masses of the two spheres:
Mass1 : Mass2 = 1 : 5
Since the mass of the second sphere is five times greater, we can say:
Mass2 = 5 * Mass1
We know that the mass is directly proportional to the volume. In other words:
Mass1 : Mass2 = Volume1 : Volume2
We can rewrite this proportion using the formula for the volume of a sphere:
(4/3) * π * (4.50 cm)^3 : Volume2 = 1 : 5
Now, we can solve for Volume2:
Volume2 = (Volume1 * 5) / 1
Finally, we can find the radius of the second sphere by rearranging the formula for the volume of a sphere and solving for the radius:
(4/3) * π * radius2^3 = Volume2
Solving this equation will give us the radius2 of the second sphere.