Question

1. Lumyn Electronics is currently renting their store space from Quality Holdings Corp.

Quality Holdings owns a total of 250 commercial properties available for rent. The company's monthly profit, \(P\), in dollars, is given by the equation

\(P(n)=-8 n^{2}+3,200 n-80,000\)

where \(\mathrm{n}\) is the number of properties rented during a particular month.

a) Would renting out all of the available properties maximize Quality Holdings' profit?

b) If not, what is the optimum number of properties that the company must have rented in order to maximize its profit?

Answer 1

\(P(n)=-8 n^{2}+3200 n-80000\)

\(n\) should be between 0 to 250

\(0 \leq n \leq 250\)

Now we have maximaize profit \(P(n)\)

\(\frac{d}{d n}(P)=\frac{d}{d n}\left(-8 n^{2}+3200 n-80000\right)\)

\(P^{\prime}(n)=-16 n+3200=0\)

\(n=\frac{3200}{16}=200\)

\(P(200)=-8 * 200^{2}+3200 * 200-80000\)

\(P(200)=240,000\)

\(P(250)=-8 * 250^{2}+3200 * 250-80000\)

\(P(250)=220,000\)

(a) renting out all of the available properties is not maximize Quality Holdings' profit

(b) \(n=200\) is optimal number of properties