Marginal Analysis of Office Complex

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Lesson Parent: 
Marginal Analysis as an Instantaneous Rate of Change
Problem: 

The cost and revenue functions for an office complex are $R(r)=-0.004r^2+55.4r$ and $C(r)=42,300-12r$, respectively, where $r$ is the monthly rent. Determine the marginal revenue and cost for monthly rent of $\$3,000$ and $\$8,000$.

Answer: 

$\$31.40$ and $-\$8.60$ for rents of $\$3,000$ and $\$8,000$, respectively.

SAMPLE SOLUTION

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Solution: 

From this lesson we know that the marginal revenue and cost are given by $R'(r)$ and $C'(r)$, respectively. Each of these is a polynomial function and we know the derivative of a polynomial. Therefore,

<Sign in to see all the formulas>

At $\$3,000<Sign in to see all the formulas>R'(\<Sign in to see all the formulas>3,000)+55.4=\$31.40$. At $\$8,000<Sign in to see all the formulas>R'(\<Sign in to see all the formulas>8,000)+55.4=-\$8.60$.

Notice that somewhere between a rent of $\$3,000$ and $\$8,000$ the marginal revenue becomes negative. This reflects the fact that as the rent increases, renters will look elsewhere to rent office space and thus increase the number of vacancies in this office complex. We can find the rent where vacancies increase by setting <Sign in to see all the formulas>. This gives

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So at a rent of $\$6,925$ we will begin loosing renters and each incremental increase in the rent will mean a decrease in revenues from renters leaving the office complex.