Question #497d1

2 Answers
Aug 5, 2015

Answer:

# 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

Answer:

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%#