Science:Math Exam Resources/Courses/MATH110/April 2018/Question 05 (a)
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Question 05 (a) 

A stone is shot vertically into the air at an initial velocity of 16 m/s. Its height (in metres) above the ground after seconds is given by (a) For how many seconds does the stone rise before starting to fall back down? 
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 

We expect that the stone rises and then falls. So, find time at which has its maximum value. 
Hint 2 

An extreme value of can only occur at a critical number. 
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. By Hint 1, we find at which the height function has its maximum value. Then, since is a local maximum, it is a critical number. Note that is a polynomial and hence differentiable on . Also, is not an endpoint. Therefore, by Hint 2, we have By the power rule for derivatives, we get Solving , we obtain Considering that for , , the height is increasing on . On the other hand, for , , so that the height is decreasing on . Therefore, the desired time is . Answer: . 