Related Rates Calculus Cone problem

[Batmaniac]

I'm stuck on this question:

"A man is sipping soda through a straw from a conical cup, 15 cm deep and 8 cm in diameter at the top. When the soda is 10 cm deep, he is drinking at the rate of 20 cm^3/s. How fast is the level of the soda dropping at that time?"

So you are given height = 15 cm, radius = 4 cm, and the derivative of the volume at height = 10 cm is 20 cm^3/s. It would appear that the question is asking for the derivative of height at 10 cm.

So volume of a cone is 1/3*pi*r^2*h, meaning, the derivative of that is:

dV/dt = (2*pi*r*dr/dt*h)/3 + (dh/dt*pi*r^2)/3

The only problem is that to find dh/dt, as the question is asking, we need to know dr/dt, and I can't think of anything I could do to find it. So perhaps I went about this question the wrong way or there is something I'm not seeing. Any help or guidance would be greatly appreciated, thanks.

 

[Hootenanny]

Can you find the height in terms of the radius and angle between the base and the hypotenuse formed by the slanted side of the cone?

 

[Batmaniac]

Since I'm fairly certain the angle between the base and the hypotenuse stays constant as 'h' decreases (and thus 'r' decreases), I can use that to find the radius at h = 10, which now that I think about it, was another unknown in the initial question that I did not have, but that still doesn't help me find dr/dt.

 

[Batmaniac]

I think I know what to do now.

When the height is 15, the radius is 5, since the angle between slant and base is constant, we have similar triangles, so the sides are proportional. So 1/3h = r. So we can sub in h into the original volume equation for r and then differentiate the equation so that our differentiated equation only has the one unknown, dh/dt and we can easily solve for it.

 

[Batmaniac]

Yeah I got the answer.

Math rules.

 

[Hootenanny]

Yeah I got the answer.

Math rules.

Good, I agree

 

[Office_Shredder]

It should definitely be noted that 90% of related rates questions use maybe 3 different easy geometric identities... if you just remember them, every question will end up asking the exact same thing, just with different numbers