If given 2 spheres, if the volume of the first sphere is 3/4 and the volume of the second sphere is 36, which is the relation of the second radius to the first?
To determine the relation between the second radius and the first, we need to compare the volumes of the spheres.
The volume of a sphere is given by the formula:
V = (4/3) * π * r^3
Let's denote the radius of the first sphere as r1 and the radius of the second sphere as r2.
Given that the volume of the first sphere is 3/4, we can express it as:
(4/3) * π * r1^3 = 3/4
To find the relation between r2 and r1, we can compare the ratios of their volumes:
[(4/3) * π * r2^3] / [(4/3) * π * r1^3] = 36 / (3/4)
Simplifying both sides of the equation, we get:
r2^3 / r1^3 = 36 / (3/4)
Since we have r2^3 / r1^3, we can cancel out the cube terms by taking the cube root on both sides of the equation:
(r2 / r1)^(3/3) = (36 / (3/4))^(1/3)
This simplifies to:
r2 / r1 = (36 / (3/4))^(1/3)
Let's calculate this value:
First, simplify the fraction 36 / (3/4) by multiplying the numerator and denominator by the reciprocal of 3/4, which is 4/3:
(36 * 4) / (3 * 3) = 48
Now, let's find the cube root of 48 by raising it to the power of 1/3:
r2 / r1 = (48)^(1/3)
Using a calculator or mathematical software, we find that the cube root of 48 is approximately 3.634.
Therefore, the relation between the second radius (r2) and the first radius (r1) is:
r2 / r1 ≈ 3.634