Question

The volume of a cantaloupe is approximated by Upper V equals four thirds pi font size decreased by 5 r cubedV= 4 3π r3. The radius is growing at the rate of 0.5 cm divided by week0.5 cm/week​, at a time when the radius is 5.85.8 cm. How fast is the volume changing at that​ moment?

Answer 1

Answer:

68.445 cm³/s

Explanation:

Given:

Volume, V = $\frac{4}{3}\pi r^3$

radius = 5.85 cm

Growth rate of radius = 0.5 cm/week

now

differentiating the volume with respect to time 't', we get:

$\frac{dV}{dt}=\frac{d(\frac{4}{3}\pi r^3)}{dt}$

or

$\frac{dV}{dt}=(\frac{4}{3}\pi )3r^2\frac{dr}{dt}$

now, substituting the value of r (i.e at r = 5.85cm) in the above equation, we get:

$\frac{dV}{dt}=4\pi 5.85^2\times 0.5$

or

$\frac{dV}{dt}=68.445cm^3/s$

hence, the rate of change of volume at r = 5.85cm is 68.445 cm³/s