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 ? µm^2
(ii) 2 to 3 µm ? µm^2
(iii) 2 to 2.1 µm ? µm^2
To find the average rate of change of V with respect to r, we need to calculate the change in V and divide it by the change in r.
(a) To find the average rate of change when r changes from 2 to 5 µm, we can use the formula:
Average rate of change = (V2 - V1) / (r2 - r1)
We need to calculate the volume at r = 2 µm and r = 5 µm:
V1 = (4/3)π(2^3) = (4/3)π(8)
V2 = (4/3)π(5^3) = (4/3)π(125)
Substituting the values:
Average rate of change = [(4/3)π(125) - (4/3)π(8)] / (5 - 2) = [4/3 π (125 - 8)] / 3 = (4/9)π(117)
(i) The average rate of change of V with respect to r when r changes from 2 to 5 µm is (4/9)π(117) µm^2.
To evaluate numerically, we can calculate:
(4/9)π(117) ≈ 163.362 µm^3
(ii) Similarly, to find the average rate of change when r changes from 2 to 3 µm, we can use the same formula:
V1 = (4/3)π(2^3) = (4/3)π(8)
V2 = (4/3)π(3^3) = (4/3)π(27)
Substituting the values:
Average rate of change = [(4/3)π(27) - (4/3)π(8)] / (3 - 2) = (4/3)π(27 - 8) = (4/3)π(19)
(iii) The average rate of change of V with respect to r when r changes from 2 to 3 µm is (4/3)π(19) µm^2.
To evaluate numerically, we can calculate:
(4/3)π(19) ≈ 79.587 µm^3
Therefore, the answers are:
(i) 163.362 µm^3
(ii) 79.587 µm^3