Science:Math Exam Resources/Courses/MATH104/December 2012/Question 04 (c)
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Question 04 (c) 

The demand equation for a certain product is where q is the number of units per hour the manufacturer can sell at a price of p dollars per unit. At what rate (in dollars per hour) must the price above ($4) be changed at this instant so that the revenue changes at 0.15 dollars per hour? 
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 

Here we are given that
as well as and from computations in the first part, we know that . We seek to find . What's the relationship between revenue, price and quantity? 
Hint 2 

Recall that . Use implicit differentiation with respect to the variable t to get the quantities we need. Notice however that we have two unknowns in our equation. Where can we find another equation? 
Hint 3 

Look back to the original given equation and differentiate implicitly with respect to the variable t. Then solve the subsequent system of equations. 
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. Proceeding as in the hints, recall that and thus Plugging in and and gives Notice that we have two unknowns and one equation so we'll need to find another equation to get our target value of . This means we need to eliminate . To do this we can differentiate the original demand equation given in the problem with respect to t, Plugging in the information above gives and simplifying This gives us our second equation. Subtracting the second equation from the first gives and thus So the rate in price must be changed by a decrease of 5 cents (0.05 dollars) per hour completing the question. 