Science:Math Exam Resources/Courses/MATH104/December 2012/Question 05 (a)
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Question 05 (a) 

Alice is 5 ft tall and Bob is 6 ft tall. At some instant they are both walking towards a street lamp 15 ft high with Alice west of the lamp and Bob east of the lamp. Suppose at this instant, Alice is walking at 4 ft/s and the distance between the tips of their shadows is decreasing at 11 ft/s. Draw a clear diagram for the situation. Let x be the distance between Alice and the lamp, and let y the distance between Bob and the lamp (you may label other distances in your diagram as well). 
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. 
Hint 1 

Each person is walking towards the lamppost, one is to the left of it and one is to the right. Where should we place the lamppost? Who is on the left? Who is on the right? 
Hint 2 

We are given that the distance between the tips of the shadow is decreasing. Consider labelling the distance of each person's shadow tip to the lamp. What is the total length between the tips? 
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.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. Consider a 15ft tall lamp at the origin of a grid (so that everyone to the right is a positive distance away and everyone to the left is a negative distance away). Let x be the distance that Alice is to the left of the lamp. We know that she is 5ft tall (which is shorter than the lamp post). Next, let y be the distance that Bob is to the right of the lamp. He is 6ft tall and so he is also shorter than the lamp but slightly taller than Alice. Rays of light will shine from the lamp, touch Alice and Bob and eventually reach the ground (forming a big triangle with the lamp and ground). Label the distance between where the light hits the ground for Alice and the lamp as C and for Bob, D. With this notation then the length of Alice's shadow, for example, should be Cx. To get a better understanding of how we could see the geometry of the problem better and how to go about solving it, it is often useful to simplify the diagram. To do this we will replace our characters with sticks since their dimensions don't affect the problem as far as we are concerned. An example of what a more mathematicallyready diagram should look like is below: The distance between the tips of the shadows is C+D and since we know this is decreasing at 11ft/s, we could write 