Related Rates: Particle Moving Along y=x^1/3, x=8, Theta Approx. 0.24 Rad

[NIZBIT]

A particle is moving along the graph of y=x^1/3.  
Suppose x is increasing at the rate of 3 cm/s.  
At what rate is the angle of inclination, theta, changing when x=8?  
[Hint: when x=8, theta approx. 0.24 rad]

I'm stuck on problem. I know x=8, y=2 and the hyp=2.87. They want the derivative of theta so am I supposed to take the derivative of theta=0.24?

 

[daveb]

No, you want to find some function that defines theta, and take the derivative of that.

 

[HallsofIvy]

A particle is moving along the graph of y=x^1/3.  
Suppose x is increasing at the rate of 3 cm/s.  
At what rate is the angle of inclination, theta, changing when x=8?  
[Hint: when x=8, theta approx. 0.24 rad]

I'm stuck on problem. I know x=8, y=2 and the hyp=2.87. They want the derivative of theta so am I supposed to take the derivative of theta=0.24?

Well, the derivative of a constant (0.24) is 0 so that wouldn't do a whole lot of good would it? It's a lot more helpful to find the derivative of a function rather than a number! Draw a picture, showing the "angle of inclination" and write theta as a function of x. Then find the derivative of that function with respect to x. (Do you remember that the derivative of a function is the tangent of the angle the tangent line to the graph makes with the x-axis?)

 

[NIZBIT]

Is this correct? Or am I completely off my rocker?

 

[quantumdude]

You're off your rocker.

Try following these steps:

1.) Draw a graph of f(x)=x^1/3 vs. x.

2.) Consider an arbitrary point on the curve: (x,y)=(x,x^1/3)

3.) Draw a line segment from the origin to (x,y) and drop a vertical line segment down from (x,y) to the x-axis. You now have a right triangle.

4.) The angle made by the line segment (the one connecting (0,0) to (x,y)) and the x-axis is the angle of inclination. Write down an expression for it in terms of x.

Try that and let's see what you come up with.

 

[NIZBIT]

Is this right?

 

[quantumdude]

So far so good, but you're supposed to plug in x=8 and get a number for d\theta/dt.

 

[NIZBIT]

Sweet! Thanks to all for the help!