Science:Math Exam Resources/Courses/MATH104/December 2010/Question 03

MATH104 December 2010

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Question 03

A city's supply of an herb comes from many small home businesses. The supply q of the herb is a function of the street price p. The Elasticity of Supply is by definition equal to $-{\frac {p}{q}}{\frac {{\textrm {d}}q}{{\textrm {d}}p}}.$ Suppose that a city's weekly supply q of the herb, in millions of grams, is related to the street price, p, in dollars per gram, by the equation $\displaystyle {}p^{3}+p=2q^{3}+q^{2}+10.$ Calculate the elasticity of supply when p=3. Please simplify. Note that when p=3, we have q=2.

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

Note that the definition of elasticity depends varies from person to person and text to text; depending on where you look, you may or may not find a minus sign in front of ${\frac {p}{q}}{\frac {dq}{dp}}.$ For this problem, we are given a definition, so we'll use this in the solution. On your exam, you might not be given the definition of elasticity. In this case, please use the definition that you were given in class. (In 2012, we did not include a minus sign in front.)

Hint 2
Use implicit differentiation to find $dq/dp$.

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. In order to find $dq/dp,$ we differentiate both sides of the supply equation with respect to $p$. $3p^{2}+1=(6q^{2}+2q){\frac {dq}{dp}}$ Solving for $dq/dp,$ ${\frac {dq}{dp}}={\frac {3p^{2}+1}{6q^{2}+2q}}$ Hence, the elasticity of supply is $-{\frac {p}{q}}{\frac {dq}{dp}}=-{\frac {p}{q}}{\frac {3p^{2}+1}{6q^{2}+2q}}.$ Evaluating this when $p=3$ and $q=2$, we obtain $-{\frac {3}{2}}{\frac {3(3)^{2}+1}{6(2)^{2}+2(2)}}=-{\frac {3}{2}}{\frac {28}{28}}=-{\frac {3}{2}}.$

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