Science:Math Exam Resources/Courses/MATH200/April 2012/Question 05 (b)

MATH200 April 2012

  •  Q1 (a)  •  Q1 (b)  •  Q2 (a)  •  Q2 (b)  •  Q3  •  Q4 (a)  •  Q4 (b)  •  Q4 (c)  •  Q5 (a)  •  Q5 (b)  •  Q6 (a)  •  Q6 (b)  •  Q7  •  Q8  •  Q9  •  Q10  •  

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  •  April 2012  •  December 2011  •  

Question 05 (b)
Let C be the intersection of the plane $\displaystyle {}x+y+z=2$ and the sphere $\displaystyle {}x^{2}+y^{2}+z^{2}=2.$ What are the coordinates of the lowest point on C?

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 hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Consider that "height" in this case is referring the value of the z coordinate. In part (a), we were trying to find the maximum value of z (i.e. the maximum height) on C. How can we use our answer from part (a) to answer this question?

Checking a solution serves two purposes: helping you if, after having used the hint, 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. From our answer in part (a), we found that there were two critical points of the objective function, $\displaystyle \Lambda (x,y,z)=f(x,y,z)+\mu (x+y+z-2)+\nu (x^{2}+y^{2}+z^{2}-2),$ on C. Those critical points corresponded to the critical points of the function ƒ(x,y,z) = z restricted to the curve C. The values of z corresponding to those critical points were z = 0 and z = 4/3. In contrast to part (a) where we looked for the larger of the two, we are now interested in the smaller value z = 0. To find the corresponding x and y value, recall the condition x = y from part (a). Plugging this into the equation for the line, we obtain $x+x+0=2\quad \rightarrow \quad x=y=1$ Therefore, the lowest point on C is (1,1,0).

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