Science:Math Exam Resources/Courses/MATH110/April 2014/Question 05

MATH110 April 2014

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Question 05
Find the coordinates of the point $P$ on the semicircle $y={\sqrt {1-x^{2}}}$ of radius $1$ (pictured below) for which the right triangle $ABP$ has maximal area.

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
Write the point $P$ as its $x$ and $y$ coordinates.

Hint 2
Define the variable that we are trying to optimise.

Hint 3
What equations might be useful in relating the variables we have?

Hint 4
Can we express the variable we are trying to optimise as a function of a single variable?

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. We can proceed by using the standard template for solving optimisation questions. The exact steps might vary between course and section, but the overall steps will be similar. 1. Let's define the coordinates of $P=(x,y)$. 2. The area of the triangle is therefore: $A={\frac {1}{2}}(1+x){\sqrt {1-x^{2}}}$ 3. The area equation is already in one variable. The domain is $x\in [-1,1]$. In that we can have $x$ between $-1$ and $1$ (the diameter of the circle) and we can include the endpoints since this will give an area of $0$. 4. Differentiate: ${\begin{aligned}A'(x)&={\frac {1}{2}}\left[(1+x)\cdot {\frac {-2x}{2{\sqrt {1-x^{2}}}}}+1\cdot {\sqrt {1-x^{2}}}\right]\\&={\frac {1}{2}}\left[{\frac {-x-x^{2}+(1-x^{2})}{\sqrt {1-x^{2}}}}\right]\\&={\frac {1}{2}}\left[{\frac {1-x-2x^{2}}{\sqrt {1-x^{2}}}}\right]\end{aligned}}$ In this case, the critical points are when the numerator is equal to zero (denominator is zero when $x=\pm 1$). ${\begin{aligned}x&={\frac {1\pm {\sqrt {1-4(-2)(1)}}}{-4}}\\&={\frac {1\pm {\sqrt {9}}}{-4}}={\frac {1}{2}},-1\end{aligned}}$ We can then use the closed interval method: ${\begin{aligned}A(1)&=0\\A(-1)&=0\\A\left({\frac {1}{2}}\right)&={\frac {1}{2}}\left(1+{\frac {1}{2}}\right){\sqrt {1-\left({\frac {1}{2}}\right)^{2}}}={\frac {3{\sqrt {3}}}{8}}>0\end{aligned}}$ 5. The maximum area happens at point $P=\left({\frac {1}{2}},{\frac {\sqrt {3}}{2}}\right)$.

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