The volume of a growing spherical cell is given below, where the radius is measured in micrometers (1 µm = 10^-6m).

V = 4/3pi r^3
(a) Find the average rate of change of V with respect to r when r changes from 2. Evaluate your answers numerically.
(i) 2 to 5 µm ? µm2
(ii) 2 to 3 µm ? µm2
(iii) 2 to 2.1 µm ? µm2

it got cut off...here's the rest:

(i) 2 to 5 µm ? µm2
(ii) 2 to 3 µm ? µm2
(iii) 2 to 2.1 µm ? µm2

Thank you

nevermind i got it

To find the average rate of change of V (the volume) with respect to r (the radius), we need to calculate the difference in the volume when the radius changes and then divide it by the difference in the radius.

We are given the formula for the volume of a sphere: V = (4/3)πr^3.

(a) (i) To find the average rate of change of V when r changes from 2 to 5 µm:
Step 1: Calculate the volume when r = 2 µm: V1 = (4/3)π(2^3) = (4/3)π(8) = (32/3)π µm^3.
Step 2: Calculate the volume when r = 5 µm: V2 = (4/3)π(5^3) = (4/3)π(125) = (500/3)π µm^3.
Step 3: Calculate the difference in volume: ∆V = V2 - V1 = (500/3)π - (32/3)π = (468/3)π = 156π µm^3.
Step 4: Calculate the difference in radius: ∆r = 5 - 2 = 3 µm.
Step 5: Calculate the average rate of change: (Average Rate of Change) = ∆V/∆r = (156π)/(3) µm^3/µm = 52π µm^2.

Therefore, the average rate of change of V with respect to r when r changes from 2 to 5 µm is 52π µm^2.

(a) (ii) To find the average rate of change of V when r changes from 2 to 3 µm:
Follow the same steps as in (a) (i) using r = 2 µm and r = 3 µm.
Step 1: V1 = (4/3)π(2^3) = (4/3)π(8) = (32/3)π µm^3.
Step 2: V2 = (4/3)π(3^3) = (4/3)π(27) = (36)π µm^3.
Step 3: ∆V = V2 - V1 = (36)π - (32/3)π = (108/3)π - (32/3)π = (76/3)π µm^3.
Step 4: ∆r = 3 - 2 = 1 µm.
Step 5: (Average Rate of Change) = ∆V/∆r = (76π)/(3) µm^3/µm = (25.33π) µm^2.

Therefore, the average rate of change of V with respect to r when r changes from 2 to 3 µm is approximately 25.33π µm^2.

(a) (iii) To find the average rate of change of V when r changes from 2 to 2.1 µm:
Follow the same steps as in (a) (i) using r = 2 µm and r = 2.1 µm.
Step 1: V1 = (4/3)π(2^3) = (4/3)π(8) = (32/3)π µm^3.
Step 2: V2 = (4/3)π(2.1^3) = (4/3)π(9.261) = (37.044/3)π µm^3.
Step 3: ∆V = V2 - V1 = (37.044/3)π - (32/3)π = (5.044/3)π µm^3.
Step 4: ∆r = 2.1 - 2 = 0.1 µm.
Step 5: (Average Rate of Change) = ∆V/∆r = (5.044π)/(3) µm^3/µm = (1.68π) µm^2.

Therefore, the average rate of change of V with respect to r when r changes from 2 to 2.1 µm is approximately 1.68π µm^2.