Science:Math Exam Resources/Courses/MATH100/December 2016/Question 11 (b)

MATH100 December 2016

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Question 11 (b)

$A$ and $B$, two people of identical height $h$, stand beneath a street lamp of height $L$. $A$ walks in a straight line and at a constant speed away from the street lamp. One second later, $B$ walks in a straight line and at the same speed, but in the opposite direction, away from the street lamp. As $A$ and $B$ move away from the lamp, their shadows grow longer. (b) As $A$ and $B$ walk away from the lamp, both their shadows are getting longer. Whose shadow is changing length faster, two seconds after $A$ left the lamp? Justify your answer.

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
First, check for which variable in the diagram we'd like to find the rate of change.

Hint 2
Use similar triangles to get rid of $a$ (or $b$ ), find $X'$ (or $Y'$), and then find $a'$ (or $b'$). Note that $x'=v=y'$, a constant speed.

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. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. The question is asking about $a'$ and $b'$. First, we compute $a'$ . From the diagram, we have $a=X-x\Rightarrow a'=X'-x'=X'-v$ , this means that first we should find $X'$: By similar triangles we have ${\begin{aligned}{\frac {h}{L}}={\frac {a}{X}}&\Rightarrow {\frac {h}{L}}={\frac {X-x}{X}}\\&\Rightarrow (1-{\frac {h}{L}})X=x\\&\Rightarrow (1-{\frac {h}{L}})X'=x'\\&\Rightarrow X'=({\frac {L}{L-h}})v\end{aligned}}$ Therefore, $a'=({\frac {L}{L-h}})v-v={\frac {h}{L-h}}v$ . Similarly, we have $b=Y-y\Rightarrow b'=Y'-y'=Y'-v$, and by similar triangles ${\begin{aligned}{\frac {h}{L}}={\frac {Y-y}{Y}}\Rightarrow (1-{\frac {h}{L}})Y=y\Rightarrow Y'=({\frac {L}{L-h}})v\end{aligned}}$ and therefore, $\color {blue}b'=({\frac {L}{L-h}})v-v={\frac {h}{L-h}}v=a'$ i.e. the rate of change in the length of shadows are equal.