Two spheres are cut from the same uniform rock. One has a radius r1 = 4.50 x 10-2 m and mass M1. The mass of the second sphere is M2 = 5M1. What is the radius of the second sphere?
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To find the radius of the second sphere, we can use the fact that the ratio of the volumes of two spheres is equal to the ratio of their radii cubed. Let's call the radius of the second sphere r2.
The volume of a sphere is given by the formula V = (4/3)πr^3.
So, the ratio of the volumes of the two spheres is:
V2/V1 = (4/3)πr2^3 / (4/3)πr1^3
V2/V1 = r2^3 / r1^3
Now, we are given that the mass of the second sphere, M2, is equal to 5 times the mass of the first sphere, M1.
The mass of a sphere is given by the formula M = density * V, where density is a constant.
Since both spheres are cut from the same uniform rock, their densities are the same, and we can write:
M2 = density * V2
M1 = density * V1
Dividing the two equations, we get:
M2/M1 = V2/V1
Substituting the values:
5M1/M1 = r2^3/r1^3
5 = r2^3/r1^3
Now, we can solve this equation for r2.
Dividing both sides by r1^3, we get:
5 = (r2/r1)^3
Taking the cube root of both sides, we get:
∛5 = ∛(r2/r1)^3
So, the radius of the second sphere, r2, is equal to the cube root of 5 times the radius of the first sphere, r1.
Therefore, r2 = ∛5 * r1
Now, substituting the given value of r1 = 4.50 x 10^-2 m, we can calculate the radius of the second sphere, r2.