a) Assuming even distribution of oil, calculate the volume in cubic meter oil slick when the radius is 1 km and the height is .23 meters
B) AT the exact instant in part a, the radius is increasing at the rate of 1.5 meters per minute. how quickly is the volume of the oil slick increasing at the same moment? How quickly is the height increasing at the same moment?
It is right cylinder
Answer:
$r_o-1000$ m
$h_o = .25$ m
Since it is even distribution,$ \frac{r_o}{h_o} = 4000 $
Thus $$r = 4000 h$$
Differentiating with respect to t,
$$\frac{dr}{dt} = 4000\frac{dh}{dt} => \frac{dh}{dt}= \dfrac{\frac{dr}{dt}}{4000}= \frac{1.5}{4000} = 0.000375\text{m/min}$$
$$V = \pi r^2 h$$
$$\frac{dV}{dt} = 2\pi r_o h_0\frac{dr}{dt}+\pi r_o^2\frac{dh}{dt} = \pi\left(2*1000*.25*1.5+1000^2*.000375\right) = 1125\pi \text{ }m^3\text{/min}$$