Science:Math Exam Resources/Courses/MATH104/December 2011/Question 05 (d)
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Question 05 (d) 

Describe geometrically (in terms of the graph of the demand curve) how you would find the price that maximizes the revenue. It might be useful to note that the slope of a line connecting a point on the demand curve to the origin is q/p. 
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 

Revenue is maximized when elasticity is 1. 
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.

Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. To maximize revenue, we need the elasticity , which is equivalent to stating Consider the line joining the origin to the point on the demand vs price curve. As the hint suggests, the slope of such a line is q/p, which is the negative of what we want to be at the price where revenue is maximized. Thus, to maximize the revenue, we can take a series of lines joining the origin to the point for different values of and determine the line such that if it is reflected across the line (so its slope is negated), it is the tangent line to the graph at the point . The slope of the tangent line is . This has a nice geometric interpretation: the line segment joining the origin to the point has same length as the line segment of the tangent line joining to the axis. In fact, an isosceles triangle is formed. Note: it is tempting to say that the revenue is maximized at if the line joining the origin to meets the curve at a right angle (recall two curves are orthogonal if the product of their slopes at their intersection point is ). But the slope of the line is and not , so this doesn't hold. We need and not . 