Science:Math Exam Resources/Courses/MATH110/April 2018/Question 09
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Question 09 

Find the dimensions of the rectangle with the largest area that can be drawn with its base on the axis and its upper corners on the parabola 
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 

Find the length of the base and height of the rectangle which has its base on the axis and its upper corners on the parabola 
Hint 2 

Find the area function of the rectangle and the range of for which the lengths of the base and height are positive. 
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Solution 

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Please rate my easiness! It's quick and helps everyone guide their studies. By drawing the graph of the parabola and the rectangle, we can see that the length of the the base is and that of the height is .
By the product rule for a derivative, the derivative of is Then, solving , we get Since is outside of the interval , we only consider . Observing that on , is increasing on . On the other hand, since on , is decreasing on . Therefore, we obtain the maximum value of the area at . Calculating the length of the base and height at , we have and . Answer: The length of the base and height are . 