Question #1683

The volume V of a growing spherical cell is given below, where the radius is measured in micrometers (1 µm = 10-6m). Evaluate your answers numerically. (Round the answers to one decimal place.)
(a) Find the average rate of change of V with respect to r when r changes from 5.
(i) 5 to 8 µm µm3 / µm
(ii) 5 to 6 µm µm3 / µm
(iii) 5 to 5.1 µm µm3 / µm

Expert's answer

The volume of the sphere is V = 4/3 π r^{3}, thus V changes

(i) from 523.6 µm3 to 2144.7 µm^{3}

(ii) from 523.6 µm3 to 904.8 µm^{3}

(iii) from 523.6 µm3 to 555.6 µm^{3}

The average rate is

(i) (2144.7+523.6)/2 = 1334.1 µm^{3}

(ii) (904.8 + 523.6) /2 = 714.2 µm^{3}

(iii) (555.6 + 523.6)/2 = 539.6 µm^{3}

(i) from 523.6 µm3 to 2144.7 µm

(ii) from 523.6 µm3 to 904.8 µm

(iii) from 523.6 µm3 to 555.6 µm

The average rate is

(i) (2144.7+523.6)/2 = 1334.1 µm

(ii) (904.8 + 523.6) /2 = 714.2 µm

(iii) (555.6 + 523.6)/2 = 539.6 µm

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