miss Thao owns a small hotel with 30 identical rooms. she finds that if she charges a price of 12 dollars ( or less) per room per day, all the rooms are rented. for every 1 dollar in price, 2 rooms remain vacant. each day, maintenance cost 4 dollars per rented room and 1 for unrented room. what price should miss thao charge n order to maximize her daily profit? hint: profit = revenue - costs
Help me, thanks a lots
current number of rooms = 30
current price per room = 12
let the number of additional increases of $1 be n
numbers of rooms rented = 30 - 2n
number of rooms not rented = 30 - (30-2n) = 2n
price per room = 12 + n
Profit = (12+n)(30-2n) - 4(30-n) - 1(2n)
= 360 + 6n -2n^2 - 120 + 4n - 2n
= -2n^2 + 8n + 240
by Calculus:
d(profit)/dn = -4n + 8
= 0 for a max of profit
4n = 8
n = 2
the price per room should be 12+n = 12+2
= $ 14.00
if you don't know calculus:
profit = -2n^2 + 8n + 240
the n of the vertex = -8/-4 = 2
so the price should be $ 14.0
tks Reiny so much :x
To find the price that maximizes Miss Thao's daily profit, we need to analyze the revenue and cost based on different prices she can charge.
Let's start by finding the number of rooms rented at different prices:
- At a price of $12 or less per room per day, all the rooms are rented.
- For every $1 decrease in price, 2 rooms remain vacant.
Based on this information, we can generate a table to track the number of rented and vacant rooms at different prices:
Price | Rented Rooms | Vacant Rooms
-------|--------------|-------------
$12 | 30 | 0
$11 | 28 | 2
$10 | 26 | 4
$9 | 24 | 6
$8 | 22 | 8
$7 | 20 | 10
$6 | 18 | 12
$5 | 16 | 14
$4 | 14 | 16
$3 | 12 | 18
$2 | 10 | 20
$1 | 8 | 22
$0 | 6 | 24
Next, we'll calculate the revenue and cost at each pricing level:
Revenue = Price * Rented Rooms
Cost = ($4 * Rented Rooms) + ($1 * Vacant Rooms)
Using this information, we can calculate the daily profit for each price:
Price | Rented Rooms | Vacant Rooms | Revenue | Cost | Profit
-------|--------------|-------------------|----------|------|--------
$12 | 30 | 0 | $360 | $120 | $240
$11 | 28 | 2 | $308 | $116 | $192
$10 | 26 | 4 | $260 | $112 | $148
$9 | 24 | 6 | $216 | $108 | $108
$8 | 22 | 8 | $176 | $104 | $72
$7 | 20 | 10 | $140 | $100 | $40
$6 | 18 | 12 | $108 | $96 | $12
$5 | 16 | 14 | $80 | $92 | -$12
$4 | 14 | 16 | $56 | $88 | -$32
$3 | 12 | 18 | $36 | $84 | -$48
$2 | 10 | 20 | $20 | $80 | -$60
$1 | 8 | 22 | $8 | $76 | -$68
$0 | 6 | 24 | $0 | $72 | -$72
From this analysis, we can see that the price that maximizes Miss Thao's daily profit is $9. At this price, she will have a daily profit of $108.
Remember, profit is calculated as Revenue - Cost.
Therefore, by charging $9 per room per day, Miss Thao will maximize her daily profit.