Hand in full solutions to the questions below. Make sure you justify all your work and include complete arguments and explanations. Your asnwers must be clear and neatly written, as well as legible (no tiny drawings or micro-handwriting please!). Your answers must be stapled, with your name and student number at the top of each page.
1. On the same page where you found this document there is a link to the worksheet ”A cylindrical tank”. Complete the worksheet and hand it in with your other work. You can attach a print out of the worksheet or copy the worksheet on your own paper.
2. Read the following problem and answer the questions below: “A rocket travels vertically from a launch pad 10 km away from an observer with a telescope. At a certain moment the angle between the telescope and the ground is π/ 3 and it is changing at a rate of 0.5 rad/s. What is the rocket’s velocity
(in km/s) at that moment?”
(a) Let x be the distance between the observer and the launch pad, y be the height of the rocket at time t , θ the angle between the telescope and the ground. Which of these variables change with time and which ones do not change with time?
(b) Using the notation introduced in part (a), write down what the problem is asking for.
(c) Using the notation introduced in part (a), explain why the following equation is true at any t : y = x tan θ , and explain why it is useful to write down such equation to solve the problem.
(d) Explain why, given the equation in part (c), sec θ =
1 cos θ
.
dy dt
= x sec 2 θ dθ dt
. Recall the definition of secant is
(e) What is the rocket’s velocity (in km/s) when the angle between the telescope and the ground is
π/ 3 and it is changing at a rate of 0.5 rad/s?
3. A Ferris wheel with a radius of 10 m is rotating at a rate of one revolution every 2 minutes. How fast is a rider rising when his seat is 16 m above the ground (assume the seat was at ground level when he got on board)?
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