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

MATH104 December 2012

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Question 04 (c)
The demand equation for a certain product is $\displaystyle {}q^{2}+p^{3/2}+2p=20$ 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 ${\frac {dR}{dt}}=0.15$ as well as $p=4$ and from computations in the first part, we know that $q=2$. We seek to find ${\frac {dp}{dt}}$. What's the relationship between revenue, price and quantity?

Hint 2
Recall that $R=pq$. 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. If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work. If you want to check your work: Don't only focus on the answer, problems are mostly marked for the work you do, make sure you understand all the steps that were required to complete the problem and see if you made mistakes or forgot some aspects. Your goal is to check that your mental process was correct, not only the result.

Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Proceeding as in the hints, recall that $R=pq$ and thus ${\frac {dR}{dt}}=q{\frac {dp}{dt}}+p{\frac {dq}{dt}}$ Plugging in ${\frac {dR}{dt}}=0.15$ and $p=4$ and $q=2$ gives $0.15=2{\frac {dp}{dt}}+4{\frac {dq}{dt}}$ Notice that we have two unknowns and one equation so we'll need to find another equation to get our target value of ${\frac {dp}{dt}}$. This means we need to eliminate ${\frac {dq}{dt}}$. To do this we can differentiate the original demand equation given in the problem with respect to t, ${\begin{aligned}{\frac {d}{dt}}\left(q^{2}+p^{3/2}+2p\right)&={\frac {d}{dt}}20\\2q{\frac {dq}{dt}}+{\frac {3{\sqrt {p}}}{2}}{\frac {dp}{dt}}+2{\frac {dp}{dt}}&=0.\end{aligned}}$ Plugging in the information above gives $2(2){\frac {dq}{dt}}+{\frac {3{\sqrt {4}}}{2}}{\frac {dp}{dt}}+2{\frac {dp}{dt}}=0$ and simplifying $4{\frac {dq}{dt}}+5{\frac {dp}{dt}}=0$ This gives us our second equation. Subtracting the second equation from the first gives $3{\frac {dp}{dt}}=-0.15$ and thus ${\frac {dp}{dt}}=-0.05$ So the rate in price must be changed by a decrease of 5 cents (0.05 dollars) per hour completing the question.

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