Question

Calc I Question!

Answer 1

Indiana Jones is trapped in an ancient enclosed conical reservoir that stands point down and has a height of 10 ft and a base radius of 5 ft. water runs in to the reservoir at a rate of 9 ft^3/min.

a.) How fast is the water level rising when the water is 6 ft deep?

b.) How much time does indy have to escape before the reservoir is completely full of water?

Answer 2

Related rates

Answer 3

here is our cone, the volume of which we know to be\[V=\frac13\pi r^2h\]we can use the relationship of similar triangles that form when the cone is partway full to put this formula in terms of one variable...

Answer 4

by similar triangles\[\frac rh=\frac5{10}\to r=\frac h2\]plugging this into our volume formula this becomes\[V=\frac13\pi (\frac h2)^2h=\frac1{12}\pi h^3\]differentiate with respect to time...

Answer 5

\[\frac{dV}{dt}=\frac14\pi h^2\frac{dh}{dt}\]we are given that dV/dt=9, because that is the rate at which the water is flowing into the cone, so we just need to solve the above for dh/dt, which is the rate change in the height of the water...

Answer 6

...and plug in h=5 because we want the dh/dt at that point

Answer 7

as for when it's full, that's pretty easy
all you need is the volume of the whole cone (of which the measurements are given) and the fact that the water is flowing in at dV/dt=9
so the time it takes to fill the tank is the total volume of the tank divided by the rate of water inflow
that part is barely even calculus!

Answer 8

although, if you want how much MORE time Indy has left after the water is at height h=5, you need to find the volume of the remaining part cone that is not filled with water:

Remaining Volume=Total volume of cone-Volume of cone filled with water

(i.e. subtract the volume of the 5 foot high cone of water from the volume of the entire cone)

then just divide the remaining volume by the rate of water flow