Science:Math Exam Resources/Courses/MATH100 A/December 2023/Question 19
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Question 19 

A rectangle is inscribed with its base on the axis and its upper corners on the parabola , as shown. What is the height of such a rectangle with the greatest possible 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 

This is an optimization problem. Can you write down a 1variable formula for the area of a rectangle that is inscribed in the region between the parabola and axis? 
Hint 2 

Let be a rectangle that is inscribed in the region between the axis and the parabola. We may as well parametrize the rectangle by the coordinate of its bottomright corner. Let us call this coordinate . Then, the topright corner of the rectangle lies on the parabola, so its coordinate is . Thus, the area of the rectangle is given by

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. We optimize the area function from the hint above. First, its derivative is given by
Setting , we find , which holds for . Since is the coordinate of the bottomright corner, it is positive, so the only extremum of occurs at . To perform our due diligence, we should check that this is indeed a maximum by computing the value . We have so . Thus, the rectangle with maximal area has its bottomright corner at the point . The height of this rectangle is . 