Question

Question #497d1

Answer 1

`111/250 pi r^3`

Explanation:

`V = 4/3 pi r^3`
`=> (dV)/(dr) = 4/cancel3 pi cancel3 r^2 = 4 pi r^2`

We know that the change in `r` is `10%`, so `delta r = r/10`.

Therefore, as a linear approximation, we may use:
`delta V = 4 pi r^2 delta r = 2 cancel4 pi r^2 r/(5 cancel10) = 2/5 pi r^3`.

Of course, you may also calculate the exact increase by substitution of `r + delta r` into the formula directly.

`tildeV = 4/3 pi (r + delta r)^3 = 4/3 pi r^3 + 4/3 pi (3 r^2 delta r + 3 r (delta r)^2 + (delta r)^3 )`
`=> tildeV = V_0 + 2 cancel4 pi r^2 r/(5 cancel10) + cancel4 pi r r^2/(25 cancel100) + cancel4/3 pi r^3/(250 cancel1000)`
`=> delta V = (2/5 + 1/25 + 1/250) pi r^3 = 111/250 pi r^3`
Of course, this is very close to the approximation we get using the differential.

Answer 2

The amount of the increase is: `0.331` times the original volume

The proportion or percent increase is: `0.331 = 33.1%`

Explanation:

`V_1 = 4/3pir^3`

When the radius increases `10%`, the new radius is `1.1r` so the new volume is:

`V_2 = 4/3pi(1.1r)^3 = 4/3pi(1.331)r^3`

`= 1.331(4/3pir^3) = 1.331V_1`

The amount of the increase is:

`V_2 - V_1 = 0.331V_1`

The proportion or percent increase is:

`(V_2-V_1)/V_1 = 0.331 = 33.1%`