Given two spheres, if the volume of the first sphere is 36r, and the volume of the second sphere is 288r, what is the relation of the second radius to the first radius?
V1 = (4pi/3)r^3 = 36r,
Divide both sides by r:
(4pi/3)r^2 = 36,
Multiply both sides by 3/4pi:
r^2 = 36 * 3/4pi,
r^2 = 27/pi = 8.59,
r = 2.93.
V2 = (4pi/3)r^3 = 288r,
(4pi/3)r^2 = 288,
r^2 = 288(3/4pi) = 216/pi = 68.75,
r = 8.29.
r2/r1 = 8.29/2.93 = 2.83,
r2 = 2.83r1.
To find the relation between the radii of two spheres when given their volumes, we can use the formula:
V = (4/3) * π * r^3
Let's translate the given information into equations.
The volume of the first sphere is given as 36r. So we have:
36r = (4/3) * π * r1^3 -- Equation 1
Similarly, the volume of the second sphere is given as 288r. So we have:
288r = (4/3) * π * r2^3 -- Equation 2
Now, we need to find the relation between r1 and r2.
To simplify the equations, let's cancel out the common factors. Divide both sides of Equation 1 by (4/3) * π:
36r / ((4/3) * π) = r1^3
Simplify:
27r / π = r1^3
Similarly, divide both sides of Equation 2 by (4/3) * π:
288r / ((4/3) * π) = r2^3
Simplify:
216r / π = r2^3
Now, we have:
r1^3 = 27r / π -- Equation 3
r2^3 = 216r / π -- Equation 4
To find the relation between r1 and r2, we can take the cube root of both equations:
r1 = (27r / π)^(1/3)
r2 = (216r / π)^(1/3)
The relation between the second radius (r2) to the first radius (r1) is:
r2 / r1 = [(216r/π)^(1/3)] / [(27r/π)^(1/3)]
To simplify the expression, we can cancel out the common terms:
r2 / r1 = [(216r/π) / (27r/π)]^(1/3)
r2 / r1 = (216r/π) / (27r/π)
r2 / r1 = 8
Therefore, the relation is that the second radius (r2) is 8 times the first radius (r1).