Question 03 (a)
A bee is flying along the curve of intersection of the surfaces and in the direction for which z is increasing. At time t = 2, the bee passes through the point (1,1,0) at speed 6.
(a) Find the velocity (vector) of the bee at time t = 2.
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
If you are stuck, check the hints below. Read the first one and consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it! If after a while you are still stuck, go for the next hint.
Write the velocity vector of the bee as
for the time-dependent space-functions and use that satisfies
The derivatives of must satisfy
Write the velocity vector in terms of the derivative you chose (lets say ), and use to calculate the value of .
Checking a solution serves two purposes: helping you if, after having used all the hints, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
We let the vector be the velocity vector. If the bee travels along a path in given by , then is given by
We are told that the bee is travelling along a path defined by the intersection of the two surfaces: and . Therefore, we can differentiate each relationship with respect to t to get a pair of equations describing the velocity of the bee at any point along the curve:
We are interested in analyzing the velocity at (1,1,0), so plug this point into the above equations:
Solving the two equations to get in terms of gives:
So at the point (1,1,0), we can write the velocity vector in terms of :
We know that the speed of the bee is 6 at the point (1,1,0), so we use the equation
Solving for gives
Taking the positive solution, (since the bee is travelling in the direction of increasing z, we can write the velocity vector