# Demand for rooms, of a hotel which has 58 rooms, is a function of price charged given by u(p)=p^2-12p+45. Find out at what price the revenue is maximized and what is the revenue?

##### 1 Answer
Jan 26, 2017

There is no limit to revenue for $p > 13$, but demand cannot be fulfilled beyond $58$ rooms, which is for $p = 13$ and maximum revenue at this level is $754$.

#### Explanation:

As the number of rooms that will be occupied, based on the price being charged, is $u \left(p\right) = {p}^{2} - 12 p + 45$ with $u \left(p\right) \le 58$ and $u \left(p\right) \in I$.

As such revenue $r$ will be given by $p \left({p}^{2} - 12 p + 45\right)$ and this will be maximized when $\frac{d}{\mathrm{dp}} r \left(p\right) = 0$, where $r \left(p\right) = {p}^{3} - 12 {p}^{2} + 45 p$ and second derivative ${d}^{2} / {\left(\mathrm{dp}\right)}^{2} r \left(p\right) < 0$ for maxima and ${d}^{2} / {\left(\mathrm{dp}\right)}^{2} r \left(p\right) > 0$ for minima.

As $\frac{d}{\mathrm{dp}} r \left(p\right) = 3 {p}^{2} - 24 p + 45$ and $3 {p}^{2} - 24 p + 45 = 0$ and dividing each term by $3$, we get

${p}^{2} - 8 p + 15 = 0$ i.e. $\left(p - 5\right) \left(p - 3\right) = 0$

and as ${d}^{2} / {\left(\mathrm{dp}\right)}^{2} r \left(p\right) = 6 p - 24 = 6 \left(p - 4\right)$

while for $p = 5$, ${d}^{2} / {\left(\mathrm{dp}\right)}^{2} r \left(p\right) = 6$

for $p = 3$, ${d}^{2} / {\left(\mathrm{dp}\right)}^{2} r \left(p\right) = - 6$

Hence, we have a local maxima at $p = 3$ and a local minima at $x = 5$

At $p = 3$, we have $r \left(3\right) = {3}^{3} - 12 \times {3}^{2} + 45 \times 3 = 54$ and at $p = 5$, we have $u \left(5\right) = {5}^{3} - 12 \times {5}^{2} + 45 \times 5 = 50$, but the latter is a local maxima and as subsequently $r \left(p\right)$ continues to rise and is limited only by $u \left(p\right) \le 58$.

And at $u \left(p\right) = 58$ and ${p}^{2} - 12 p + 45 = 58$ is ${p}^{2} - 12 p - 13 = 0$ i.e. $\left(p - 13\right) \left(p + 1\right) = 0$

Revenue is maximized at $p = 13$, where it is

${13}^{3} - 12 \times {13}^{2} + 45 \times 13 = 2197 - 2028 + 585 = 754$, when occupancy is $58$.

However, as even beyond $p = 13$ i.e. for $p > 13$, demand for rooms continues to increase. Hence, answer is that there is no limit post the price $p > 13$,
graph{x^3-12x^2+45x [-5, 15, -50, 800]}