Question #497d1

2 Answers
Aug 5, 2015

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.

Aug 5, 2015

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%