Science:Math Exam Resources/Courses/MATH110/April 2018/Question 06 (a)
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Question 06 (a) 

Read the problem below and answer the questions in part (a) and (b). Makes sure your solution includes a sketch labeled consistently with variables in your calculations. Your asnwers should be a numerical value, but you do not need to simplify it. A small spider is crawling along the graph of a parabola in the first quadrant (where and are measured in cm) in such a way that its coordinate increases at a constant rate of . The spider is pulling a thin thread of silk with it that is fixed at the origin. (a) When the spider is at the point , how fast is the silk thread lengthening? 
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 relation between the length of the silk thread and the coordinate of the location of the spider. 
Hint 2 

Denoting the coordinate of the location of the spider by , we have . 
Hint 3 

Use either the chain rule (Solution 1) or the implicit differentiation (Solution 2). 
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 1 

Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. Let be the coordinate of the location of the spider in time . Since the spider moves along the graph , the location of the spider in time is . Note that the length of the silk thread is the distance between two points and . Using the distance formula, we get Since is a function of and is the function of , by the chain rule, we have By the Hint 2, we have . Since can be written as the composite of two functions , where and , we apply the chain rule with and to get Plugging and into the formula for , we obtain At the location of the spider , i.e., , we therefore have Answer: 
Solution 2 

Found a typo? Is this solution unclear? Let us know here.
Please rate my easiness! It's quick and helps everyone guide their studies. Let be the coordinate of the location of the spider in time . Since the spider moves along the graph , the location of the spider in time is . Note that the length of the silk thread is the distance between two points and . Using the distance formula, we get Now, we find using the implicit differentiation. Since we take the derivative with respect to to get
On the other hand, we compute the derivative of the function with respect to using chain rule (note that x depends on time t):
Therefore
Noting that and , we have Finally we isolate the derivative of with respect to to get
