You and your roommate have timed the necessary physical operations. The ﬁrst physical production step is to wash out the mixing bowl from the previous batch, add all of the ingredients, and mix them in your food processor. The mixing bowls hold ingredients for up to three dozen cookies. You then dish up the cookies, one dozen at a time, onto a cookie tray. These activities take six minutes for the washing and mixing steps, regardless of how many cookies are being made in the batch. That is, to mix enough dough and ingredients for three dozen cookies takes the same six minutes as one dozen cookies. However, dishing up the cookies onto the tray takes two minutes per tray. The next step, performed by your roommate, is to put the cookies in the oven and set the thermostat and timer, which takes about one minute. The cookies bake for the next 9 minutes. So total baking time is 10 minutes, during the ﬁrst minute of which your roommate is busy setting the oven. Because the oven only holds one tray, a second dozen takes an additional 10 minutes to bake.

Question

You and your roommate have timed the necessary physical operations. The ﬁrst physical production step is to wash out the mixing bowl from the previous batch, add all of the ingredients, and mix them in your food processor. The mixing bowls hold ingredients for up to three dozen cookies. You then dish up the cookies, one dozen at a time, onto a cookie tray. These activities take six minutes for the washing and mixing steps, regardless of how many cookies are being made in the batch. That is, to mix enough dough and ingredients for three dozen cookies takes the same six minutes as one dozen cookies. However, dishing up the cookies onto the tray takes two minutes per tray. The next step, performed by your roommate, is to put the cookies in the oven and set the thermostat and timer, which takes about one minute. The cookies bake for the next 9 minutes. So total baking time is 10 minutes, during the ﬁrst minute of which your roommate is busy setting the oven. Because the oven only holds one tray, a second dozen takes an additional 10 minutes to bake.

Your roommate also performs the last steps of the process by ﬁrst removing the cookies from the oven and putting them aside to cool for ﬁve minutes, then carefully packing them in a box and accepting payment. Removing the cookies from the oven takes only a negligible amount of time, but it must be done promptly. It takes two minutes to pack each dozen and about one minute to accept payment for the order.

How many orders can you fill in a night, assuming you are open four hours each night?

Answer 1

To calculate how many orders you can fill in a night, we need to consider the time it takes for each step and the total time available.

First, let's calculate the time it takes to make one batch of cookies:
- Washing and mixing: 6 minutes
- Dishing up onto trays: 2 minutes
- Putting cookies in the oven: 1 minute
- Baking time: 10 minutes
- Cooling time: 5 minutes
- Packing one dozen: 2 minutes
- Accepting payment: 1 minute

Total time for one batch of cookies: 6 + 2 + 1 + 10 + 5 + 2 + 1 = 27 minutes

Now, let's calculate the total time available in a night:
4 hours = 4 * 60 = 240 minutes

To determine how many orders can be filled in a night, we divide the total time available by the time it takes for one batch:
240 minutes / 27 minutes = 8.89 batches

Since you can only make full batches of cookies, the maximum number of orders you can fill in a night is 8 orders.

Answer 2

To determine the number of orders that can be filled in a night, we need to calculate the total time required for each order and divide the available time by that value.

Let's break down the time required for each step:

1. Washing and mixing: 6 minutes (regardless of the number of cookies being made)
2. Dishing up cookies onto trays: 2 minutes per tray
3. Setting the oven: 1 minute
4. Baking time: 10 minutes for the first dozen, additional 10 minutes for each additional dozen
5. Cooling time: 5 minutes
6. Packing each dozen: 2 minutes per dozen
7. Accepting payment: 1 minute

Now, let's calculate the time required for a single order:

- Washing and mixing: 6 minutes
- Dishing up cookies: 2 minutes per tray (assuming one tray for each dozen)
- Setting the oven: 1 minute
- Baking time: 10 minutes for the first dozen, additional 10 minutes for each additional dozen (assuming one dozen per tray)
- Cooling time: 5 minutes
- Packing: 2 minutes per dozen
- Accepting payment: 1 minute

Total time per order = 6 + 2 + 1 + 10 + 5 + 2 + 1 = 27 minutes

Now, let's calculate the available time in a night:

As stated, you are open for 4 hours each night. Since 1 hour is equal to 60 minutes, the total available time in minutes is 4 * 60 = 240 minutes.

Finally, we can determine the number of orders that can be filled:

Number of orders = Total available time / Time per order
Number of orders = 240 minutes / 27 minutes per order

Calculating this, we find that you can fill approximately 8.89 orders in a night.

Since we cannot have a fraction of an order, the maximum number of orders you can fill in a night is 8.

Therefore, assuming you are open for four hours each night, you can fill up to 8 orders.