# Thread: Rod and Wheel Calculus question

1. ## Rod and Wheel Calculus question

Ok so I am struggling with a question that I have been trying to figure out for a couple of days now. The professor told us the first part but i can't figure out the rest. Here is the question from the attached files:

According to our professor, question 1 becomes:

X(t) = 2[cos(7πt) + sqrt(9-sin2(7πt))]

Jest before someone asks, it is not graded HWK. It is an application problem which encompasses many of the things we have learnt in the last week. We get them every week and must try to get them (not necessary though).

2. What exactly is your problem? I got the same result for x(t). The angular speed is (7/2) * (2π radians) / (1 min) = 7π radians/min. Therefore, the angle the radius to point A makes at time t is 7πt radians. Next, x(t) is the sum of the projection of A, i.e., 2cos(7πt), and the projection of the rod, which can be found using Pythagoras theorem (the y-coordinate of A is 2sin(7πt)).

The derivative can be found using the chain rule. It's a big expression, but sin(7πt) can be factored out. One can show that the second factor cannot equal zero. Therefore, x'(t) = 0 iff sin(7πt) = 0.

3. ok... how do I go about proving that x'(t) = 0? It asks to use Calculus and I need points.

4. I am not sure I understand your question. You can't prove that x'(t) = 0 for every t. So for what t do you want to prove it and why?

You are asked to find t such that the speed of B at time t is zero. By calculating the speed of B as the derivative x'(t), you have already used calculus. Next, you need to solve the equation x'(t) = 0. As I said, this equation is equivalent to sin(7πt) * f(t) = 0 for some function f(t). I believe it it possible to show that f(t) is never zero. Therefore, x'(t) = 0 iff sin(7πt) = 0. You should be able to solve the latter equation.

5. ok I understand... I was making it more complicated than it had to be... so the only way for sin(7πt) = 0 is if t = 0 right?

6. No, also when 7πt = π, 2π, 3π and in general kπ for some integer k.

7. right! Sorry about that and thanks for all the help! Greatly appreciated