a ball has a volume of 45. the radius of the larger ball is 3 times the small ball what is the volume?
Assume the smaller ball has a volume of 45.
Since volume is proportional to r^3, and R = 3r, the larger V is 3^3 as big as the smaller v, or 27*45 = 1215.
for the details, see below
for the smaller ball, v = kr^3 with k=4pi/3
The larger ball with radius R=3r has volume
V = kR^3 = k(3r)^3 = 27kr^3 = 27v = 27*45 = 1215
To find the volume of the larger ball, we need to find the radius of the smaller ball and use that to calculate the volume ratio.
Let's start with the information given: the volume of the smaller ball is 45 units.
The volume of a sphere is given by the formula V = (4/3)πr^3, where V is the volume and r is the radius.
For the smaller ball, we have the volume V = 45. Therefore, we can rewrite the formula as 45 = (4/3)πr^3.
Now, we need to find the radius of the smaller ball. To do that, we can rearrange the formula:
r^3 = (3/4)(45/π)
r^3 = 135/4π
r = (135/4π)^(1/3)
Next, we are given that the radius of the larger ball is 3 times the radius of the smaller ball. So, we can find the radius of the larger ball by multiplying the radius of the smaller ball by 3:
R = 3 * r
Once we have the radius of the larger ball, we can find its volume using the formula V = (4/3)πR^3:
V = (4/3)π(3r)^3
= (4/3)π27r^3
= 36πr^3
Therefore, the volume of the larger ball is 36πr^3.