Difficult Implicit derivative.

[daniel2.0]

Hey all, I am a new user, please let me know if is the wrong area. This problem is about the using the first derivative to determine a rate.

I have done most of the problem, and I am struggling with the last part. I have to take the derivative with respect to time.

To make it easier to read, I uploaded my problem into an image. The area below the red line is where I need the most help. I think I have everything above complete correctly. I will post what I got for 46.1.2. Click to expand image.




46.1.2
What is the radius at the top of the cone?
I got a radius of 9cm at the very top of the cone. 99% sure this is right.

Use the similar triangle concept (basic trig) to express r in terms of h.
Using trig (and looking at the red triangle at bottom of problem) we know that . Using the volume formula given in the problem, we could say that v=(pi/3)(htan(theta))^2h. Is this correct?


46.1.3
How do I use the formula I got to answer the first paragraph? I know I am supposed to take the derivative with respect to time, but how? I have trouble with derivatives using multiple variables. I don't know if I need this, but for theta I got .6435rad.


So, to sum it all up, I need to get the answer for the last sentence of the first paragraph. How do I go about doing that?

Thanks all! (and sorry about the wall of text)

 

[Deleted member 4993]

Hey all, I am a new user, please let me know if is the wrong area. This problem is about the using the first derivative to determine a rate.

I have done most of the problem, and I am struggling with the last part. I have to take the derivative with respect to time.

To make it easier to read, I uploaded my problem into an image. The area below the red line is where I need the most help. I think I have everything above complete correctly. I will post what I got for 46.1.2. Click to expand image.




46.1.2
What is the radius at the top of the cone?
I got a radius of 9cm at the very top of the cone. 99% sure this is right.

Incorrect.

To check, find the volume of a cone whose height is 12 cm and the radius of the top circle is 9 cm


Use the similar triangle concept (basic trig) to express r in terms of h.
Using trig (and looking at the red triangle at bottom of problem) we know that View attachment 1813. Using the volume formula given in the problem, we could say that v=(pi/3)(htan(theta))^2h. Is this correct?

It would be better to write:

V = π/3 * tan²(Θ) * h³

Remember that Θ is constant for a given cone.

46.1.3
How do I use the formula I got to answer the first paragraph? I know I am supposed to take the derivative with respect to time, but how? I have trouble with derivatives using multiple variables. I don't know if I need this, but for theta I got .6435rad.

Look at the equation above (that you derived) carefully - you have V expressed as a function of single variable - h.

Both of those are functions of time. So you can write:

V(t) = π/3 * tan²(Θ) * [h(t)]³

dV(t)/dt = π * tan²(Θ) * [h(t)]²

So, to sum it all up, I need to get the answer for the last sentence of the first paragraph. How do I go about doing that?

Thanks all! (and sorry about the wall of text)

.