The volume of a sphere is given by:
V=4/3 pie r cubed.
Use the formula, and show your workings, to find out what happens to the volume of a sphere if r is halved and r is trippled?
Please help as this has just completly lost me, i end up with
V = 4/3 Pie r cubed / eight .....for half of r.
You were correct,
when you cut the radius in half, the volume will be 1/8 of the original.
When you triple the radius, the new volume will be 27 times that of the original.
Look at what happens when we substitute the new radius for r
sub in 3r for the radius
V = (4/3)π(3r)^3
= (4/3)π)(27)r^3
= 27 * (4/3)πr^3
Notice that the volume changes by the cube of the scaling factor,
taking 1/2 ----> (1/2)^3 = 1/8
To find out what happens to the volume of a sphere when the radius is halved, we can substitute the new radius (r/2) into the formula:
V1 = (4/3) * π * (r/2)^3
Simplifying this expression:
V1 = (4/3) * π * (r^3/8)
V1 = (r^3/6) * π
So, when the radius is halved, the volume of the sphere becomes (1/6) times the volume of the original sphere.
Now, let's find out what happens to the volume of a sphere when the radius is tripled. Substituting the new radius (3r) into the formula:
V2 = (4/3) * π * (3r)^3
Simplifying this expression:
V2 = (4/3) * π * (27r^3)
V2 = 36 * (r^3/3) * π
V2 = 12 * r^3 * π
So, when the radius is tripled, the volume of the sphere becomes 12 times the volume of the original sphere.
To summarize:
- When the radius is halved, the volume of the sphere becomes (1/6) times the original volume.
- When the radius is tripled, the volume of the sphere becomes 12 times the original volume.
To find out what happens to the volume of a sphere when the radius (r) is halved and when it is tripled, we can substitute the given values into the formula V = (4/3)πr^3 and calculate the new volumes.
Let's start by considering the scenario where the radius is halved.
1. When the radius (r) is halved, the new radius will be (r/2). So we substitute r/2 into the formula and simplify:
V_1 = (4/3)π(r/2)^3
V_1 = (4/3)π(r^3/8)
V_1 = (1/6)πr^3
So when the radius is halved, the new volume (V_1) is equal to (1/6) times the original volume.
Now let's move on to the scenario where the radius is tripled.
2. When the radius (r) is tripled, the new radius will be (3r). So we substitute 3r into the formula and calculate:
V_2 = (4/3)π(3r)^3
V_2 = (4/3)π(27r^3)
V_2 = 36πr^3
Therefore, when the radius is tripled, the new volume (V_2) is equal to 36 times the original volume.
To conclude, if we decrease the radius to half its original value, the volume becomes 1/6 of the original volume. On the other hand, if we increase the radius to three times its original value, the volume becomes 36 times the original volume.