Let's break down each question and calculate the answers step by step:

$1.$ Compute the daily capacity of the Preparation operation.

The Preparation operation takes $25$ minutes per batch, and the system works for $6$ hours per day $(360$ minutes$).$ To find the daily capacity, divide the total available time by the time it takes for one batch:

Daily Capacity $=$ Total Available Time $/$ Time per Batch

Daily Capacity $=360$ minutes $/25$ minutes per batch

Daily Capacity $14\mathrm{.}4$ batches

Answer: D: $14\mathrm{.}4$ batches.

$2.$ Compute the daily capacity of the Bake operation.

The Bake operation takes $15$ minutes per batch, and the daily capacity can be calculated in the same way:

Daily Capacity $=$ Total Available Time $/$ Time per Batch

Daily Capacity $=360$ minutes $/15$ minutes per batch

Daily Capacity $=24$ batches

Answer: A: $24$ batches.

$3.$ Compute the daily capacity of the Packaging operation.

The Packaging operation takes $10$ minutes per batch, and again, we'll use the same approach:

Daily Capacity $=$ Total Available Time $/$ Time per Batch

Daily Capacity $=360$ minutes $/10$ minutes per batch

Daily Capacity $=36$ batches

Answer: C: $36$ batches.

$4.$ Compute the daily process capacity.

The processing capacity is determined by the capacity of the slowest operation, which is the Bake operation $(24$ batches$).$ Therefore, the daily process capacity is $24$ batches.

Answer: A: $24$ batches.

Explanation:

Let's go through each question step by step with clear explanations for each answer and why the other options are not correct:

$1.$ Compute the daily capacity of the Preparation operation.

To find the daily capacity of the Preparation operation, we need to calculate how many batches of muffins can be prepared in a day.

Given:

Preparation time per batch: $25$ minutes

Total available working time per day: $6$ hours $=360$ minutes

Calculation:

Daily Capacity $=$ Total Available Time $/$ Time per Batch

Daily Capacity $=360$ minutes $/25$ minutes per batch

Daily Capacity $14\mathrm{.}4$ batches

Explanation: The Preparation operation takes $25$ minutes per batch. Therefore, in a day with $360$ minutes of working time, you can prepare approximately $14\mathrm{.}4$ batches of muffins.

$2.$ Compute the daily capacity of the Bake operation.

Given:

Bake time per batch: $15$ minutes

Total available working time per day: $6$ hours $=360$ minutes

Calculation:

Daily Capacity $=$ Total Available Time $/$ Time per Batch

Daily Capacity $=360$ minutes $/15$ minutes per batch

Daily Capacity $=24$ batches

Explanation: The Bake operation takes $15$ minutes per batch. In a day with $360$ minutes of working time, you can bake $24$ batches of muffins.

$3.$ Compute the daily capacity of the Packaging operation.

Given:

Packaging time per batch: $10$ minutes

Total available working time per day: $6$ hours $=360$ minutes

Calculation:

Daily Capacity $=$ Total Available Time $/$ Time per Batch

Daily Capacity $=360$ minutes $/10$ minutes per batch

Daily Capacity $=36$ batches

Explanation: The Packaging operation takes $10$ minutes per batch. In a day with $360$ minutes of working time, you can package $36$ batches of muffins.

$4.$ Compute the daily process capacity.

The process capacity is determined by the operation with the lowest capacity. In this case, it's the Bake operation with a capacity of $24$ batches per day.

Explanation: The daily process capacity is constrained by the slowest operation, which is baking, allowing for $24$ batches of muffins to be produced per day.

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Step $2$

$5.$ Where is the bottleneck?

The bottleneck is the operation with the lowest capacity, which is the Bake operation $(24$ batches$).$

Answer: E: Bake.

$6.$ Suppose the capacity of the bottleneck is doubled. What is the new process capacity per day?

If the capacity of the Bake operation is doubled, the new daily capacity becomes $2*24=48$ batches.

Answer: B: $37$ batches.

$7.$ Where is the new bottleneck$($s$)?$

If the capacity of the Bake operation is doubled, the new bottleneck will be the next slowest operation, which is the Preparation operation.

Answer: A: Preparation.

Please note that I corrected the options for questions $6$ and $7$ based on the information provided in the previous questions and their answers.

Explanation:

$5.$ Where is the bottleneck?

The bottleneck is the operation that limits the overall process capacity. It's the operation with the lowest capacity, which in this case is the Bake operation with a capacity of $24$ batches per day.

Explanation: The Bake operation is the bottleneck because it has the lowest capacity compared to the other operations.

$6.$ Suppose the capacity of the bottleneck is doubled. What is the new process capacity per day?

If the capacity of the bottleneck operation $($Bake$)$ is doubled from $24$ to $48$ batches, the new daily pr