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Lindsay owns a parking lot with 100 spaces in downtown Portland. She can offer an "early bird" special for $12/day. She knows that she can attract as many business people as parking spots that she is willing to allocate at this low daily rate. If she doesn’t offer a space for an "early bird" special, she can rent the space to a day user who will pay $6/hour. On average, each space will be utilized for 3.5 hours per day.

Here is a historical tally on how many parking spaces she has had occupied by day users at the hourly rate:

| Parking spaces | Probability |
|----------------|-------------|
| 65 | 0.15 |
| 70 | 0.20 |
| 75 | 0.25 |
| 80 | 0.30 |
| 85 | 0.10 |

How many spaces should Lindsay allocate to the early birds to maximize her revenue?

Answer :

To maximize her revenue, Lindsay should allocate 41 spaces for the "early bird" special and the remaining 59 spaces for day users.

To determine how many spaces Lindsay should allocate to the "early bird" special to maximize her revenue, we need to compare the revenue she would generate from "early bird" customers with the revenue she would generate from day users.

Let's start by calculating the revenue generated from the "early bird" special. Since Lindsay can attract as many business people as parking spots she is willing to allocate at $12/day, the revenue from "early bird" customers would be the number of allocated spaces multiplied by the special rate.

Let's assume she allocates x spaces for the "early bird" special. The revenue from "early bird" customers would be 12x dollars.

Next, let's calculate the revenue generated from day users. We know that each parking space will be utilized for an average of 3.5 hours per day. Using the historical tally data, we can calculate the expected revenue from day users. The revenue from day users would be the sum of the products of the number of occupied spaces and the hourly rate for each corresponding probability.

Expected revenue from day users = (65 * 0.15 * 6) + (70 * 0.20 * 6) + (75 * 0.25 * 6) + (80 * 0.30 * 6) + (85 * 0.10 * 6) = 97.5 + 84 + 112.5 + 144 + 51 = 489 dollars.

Now, we need to find the optimal value for x, the number of spaces allocated for the "early bird" special, to maximize revenue. We can compare the revenue from "early bird" customers (12x) with the revenue from day users (489) and find the value of x that maximizes this expression.

12x > 489

Solving this inequality for x, we find:

x > 489 / 12

x > 40.75

Since the number of spaces cannot be fractional, Lindsay should allocate 41 spaces for the "early bird" special to maximize her revenue.

To maximize her revenue, Lindsay should allocate 41 spaces for the "early bird" special and the remaining 59 spaces for day users.

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