Problem 16E

16. (a) The volume of a growing spherical cell is $V=\frac{4}{3} \pi r^{3}$, where the radius $r$ is measured in micrometers $\left(1 \mu \mathrm{m}=10^{-6} \mathrm{~m}\right)$. Find the average rate of change of $V$ with respect to $r$ when $r$ changes from

(i) 5 to $8 \mu \mathrm{m}$

(ii) 5 to $6 \mu \mathrm{m}$

(iii) 5 to $5.1 \mu \mathrm{m}$

(b) Find the instantaneous rate of change of $V$ with respect to $r$ when $r=5 \mu \mathrm{m}$

(c) Show that the rate of change of the volume of a sphere with respect to its radius is equal to its surface area. Explain geometrically why this result is true. Argue by analogy with Exercise $13(\mathrm{c})$.

 

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