Thread: 3 related rate problems, I am truly lost.

1. 3 related rate problems, I am truly lost.

Hi, i have no idea where to begin on these three related rate problems.

A water-f lled spherical tank with a radius of 1 meter empties from a hole in the bottom in
such a way that the water level decreases at a constant rate of 3 centimeters per second. How fast
is the volume of water in the tank changing when the tank is half full?

A girl blows up a spherical balloon with a face drawn on it. She blows 100 cubic centime-
ters of air into the balloon every second. When t = 2 seconds, the eyes of the face on the balloon
are 4 centimeters apart, measured along the surface of the balloon. How far apart are the eyes after
5 seconds, measured along the surface of the balloon? You may assume that the balloon stretches
uniformly.

A flying saucer circles the earth from pole to pole at a height of 500 miles and a speed of
10,000 miles per hour. A boy lying on his back on the ice at the north pole watches it
How fast is the
flying saucer moving away from the boy when it passes over the horizon? (Hint:
you will need to know the radius of the Earth.)

You do not need to answer all of the problems if you do not wish to.
I greatly appreciate any help provided.

2. Re: 3 related rate problems, I am truly lost.

i think i got the answer to number 1, i calculated 6pi

3. Re: 3 related rate problems, I am truly lost.

Originally Posted by gosbor6
i think i got the answer to number 1, i calculated 6pi
don't think so ...

let $\displaystyle x^2 + y^2 = 1$ be the side view of the spherical tank.

let $\displaystyle h$ = water level in the tank , $\displaystyle -1 \le h \le 1$

Water volume in the tank can be modeled by the accumulation function ...

$\displaystyle V = \pi \int_{-1}^h \left(1 - y^2 \right) \, dy$

using the FTC ...

$\displaystyle \frac{dV}{dt} = \pi \left(1 - h^2 \right) \cdot \frac{dh}{dt}$

... sub in your given values and determine $\displaystyle \frac{dh}{dt}$ .

Note that these are not what I would call "basic" related rates problems ...

4. Re: 3 related rate problems, I am truly lost.

Hi skeeter, thanks for your quick response! sorry for my not so quick response i was at work today.
I am currently enrolled in calculus 1 and we have yet to start doing integrals and i was wondering if there was a simpler way to get the answer, such as:
since you want to find the rate its draining at halfway. i would just take pi r^2 and use that.
2pi r (in this case r is 1) dh/dt(3).
is there something im missing?
thank you.