Thread: Need some explanation on related rates problems.

Hello fellow Mathematicians!

I am working on some pratice problem and I have the answer sheet for it but no work. I keep getting the wrong answers so it would be great if someone can help me or at least guide me toward the steps for getting the answers. I really want to learn how to solve it. I have read the book chapter on related rates but O need to kind of wrap my head around these problems any conceptual explanation is great too! Thanks!


1.

Gravel is being dumped from a conveyor belt at a rate of 50 cubic feet per minute. It forms a pile in the shape of a right circular cone whose base diameter and height are always the same. How fast is the height of the pile increasing when the pile is 21 feet high?
Recall that the volume of a right circular cone with height h and radius of the base r is given by

The answer is 0.144

2. A spherical snowball is melting in such a way that its diameter is decreasing at rate of 0.2 cm/min. At what rate is the volume of the snowball decreasing when the diameter is 10 cm. (Note the answer is a positive number).

Answer: 31.41

So I know the formula for a sphere is
so do I plug in the r, which is .1. and then do the derivative of it? Isn't .2 cm/min the derivation already? thanks3.

3. A street light is at the top of a 18 ft tall pole. A woman 6 ft tall walks away from the pole with a speed of 8 ft/sec along a straight path. How fast is the tip of her shadow moving when she is 45 ft from the base of the pole?

Answer: 14

It said to use two triangles to compare. I am not sure how to do this problem. I feel like I am missing something here...

Thank you for the help and insight!

Originally Posted by everyonelovesmath

Hello fellow Mathematicians!

I am working on some pratice problem and I have the answer sheet for it but no work. I keep getting the wrong answers so it would be great if someone can help me or at least guide me toward the steps for getting the answers. I really want to learn how to solve it. I have read the book chapter on related rates but O need to kind of wrap my head around these problems any conceptual explanation is great too! Thanks!


1.

Gravel is being dumped from a conveyor belt at a rate of 50 cubic feet per minute. It forms a pile in the shape of a right circular cone whose base diameter and height are always the same. How fast is the height of the pile increasing when the pile is 21 feet high?
Recall that the volume of a right circular cone with height h and radius of the base r is given by

The answer is 0.144

2. A spherical snowball is melting in such a way that its diameter is decreasing at rate of 0.2 cm/min. At what rate is the volume of the snowball decreasing when the diameter is 10 cm. (Note the answer is a positive number).

Answer: 31.41

So I know the formula for a sphere is
so do I plug in the r, which is .1. and then do the derivative of it? Isn't .2 cm/min the derivation already? thanks3.

You should see immediately that it make no sense at all to "plug in the r" and then "do the derivative"! Once you "plug in the r" you no longer have a function to differentiate! Also it makes no sense to say "isn't .2 cm/min the derivation already". Certainly, as a rate of change, it is a derivative but the derivative of what? There are two varying quantities here, the volume and the radius, and so two derivatives with respect to time.

Given that A= f(B), by the chain rule, dA/dt= df/dB dB/dt. That's the whole point of every "related rates" problem. You get some "static" formula (not involving the time, t) connecting the quantities given in a problem, then differentiate with respect to t.
Yes, . What is dA/dr? So then what is dA/dt= dA/dr dr/dt? After you have that, put in the given specific values.

3. A street light is at the top of a 18 ft tall pole. A woman 6 ft tall walks away from the pole with a speed of 8 ft/sec along a straight path. How fast is the tip of her shadow moving when she is 45 ft from the base of the pole?

Answer: 14

It said to use two triangles to compare. I am not sure how to do this problem. I feel like I am missing something here...

Have you drawn a picture? Draw the pole, the woman, and the ground. Draw the path of the light from lamp to the top of the womans head to the ground (at the tip of her shadow, of course). You should see two similar right triangles, both having the ground as one leg, one having the woman as the other leg, the other triangle, the lamp post.

Thank you for the help and insight!

Hi, Thanks, I was able to get figure out questions 1 and 3, but could you elaborate on the shadow problem?


also, how would you go about doing a problem like this?

At noon, ship A is 5 nautical miles due west of ship B. Ship A is sailing west at 15 knots and ship B is sailing north at 15 knots. How fast (in knots) is the distance between the ships changing at 5 PM? (Note: 1 knot is a speed of 1 nautical mile per hour.)

Thanks!