Question

A bottling plant has 126,515 bottles with a capacity of \(355 \mathrm{~mL}\), 108,500 caps, and \(48,775 \mathrm{~L}\) of beverage. (a) How many bottles can be filled and capped? (b) How much of each item is left over? (c) Which component limits the production?

Short answer

(a) 108,500 bottles can be filled and capped. (b) Leftover quantities: 18,015 bottles, 0 caps, and 4,915 L of beverage. (c) The limiting component is the caps.

Step-by-step solution

Calculate the number of bottles that can be filled

First, let's calculate how many bottles the given quantity of beverage can fill. We are given 48,775 L of beverage, and each bottle has a capacity of 355 mL. To find the number of bottles that can be filled, we need to convert the beverage quantity to milliliters and divide the total by the capacity of each bottle. \(1 \mathrm{L} = 1000 \mathrm{mL}\) So, \(48,775 \mathrm{~L} = 48,775 \times 1000 \mathrm{~mL}\) Now, we can divide the total beverage quantity (in milliliters) by each bottle's capacity: Number of bottles filled with beverage = \(\frac{48,775 \times 1000 \mathrm{~mL}}{355 \mathrm{~mL}}\)

Calculate the number of bottles that can be capped

Now, let's calculate how many bottles can be capped using the given number of caps. Since each cap can only be used on one bottle, the maximum number of bottles that can be capped is equal to the number of caps: Number of bottles capped = 108,500

Determine the minimum number of bottles filled and capped

We can only cap bottles that have also been filled with beverage. Therefore, the number of bottles that can be both filled and capped is the lowest value between the number of bottles filled and the number of bottles capped: Minimum number of bottles filled and capped = min(\(\frac{48,775 \times 1000 \mathrm{~mL}}{355 \mathrm{~mL}}\), 108,500)

Calculate the leftover quantities

To find the leftovers of bottles, caps, and beverage, we need to subtract the number of bottles filled and capped from the total quantities: Leftover bottles: 126,515 - (Minimum number of bottles filled and capped) Leftover caps: 108,500 - (Minimum number of bottles filled and capped) Leftover beverage: \(48,775 \times 1000 \mathrm{~mL} -\)(Minimum number of bottles filled and capped)\(\times 355 \mathrm{~mL}\)

Identify the limiting component

The limiting component will be the one that can be used up completely when filling and capping the bottles. In other words, it is the item with the smallest number left after determining the number of bottles filled and capped: Limiting component = min(leftover bottles, leftover caps, leftover beverage) Now, let's find the values for our answers. (a) The number of bottles filled and capped: \(Minimum \ number \ of \ bottles \ filled \ and \ capped = min(\frac{48,775 \times 1000 \mathrm{~mL}}{355 \mathrm{~mL}}, 108,500) = min(137,464, 108,500) = 108,500\) (b) Leftover quantities: Leftover bottles = \(126,515 - 108,500 = 18,015\) Leftover caps = \(108,500 - 108,500 = 0\) Leftover beverage = \((48775\times1000-108500\times355) \mathrm{~mL} = 4,915,000 \mathrm{~mL} = 4,915 \mathrm{~L}\) (c) Limiting component: Since there are no leftover caps, the limiting component is the caps.

Key concepts

Stoichiometry and Its Role in Production

Stoichiometry is a fundamental concept in chemistry that deals with the calculation of reactants and products in chemical reactions. It's just like piecing a puzzle together, where the picture can't be completed if any pieces are missing or don't fit.
In the context of a bottling plant, stoichiometry helps us determine how many bottles can be filled with a given amount of beverage. To calculate this, we need to know the conversion rates and how they interact, for example, how liters convert to milliliters.
By employing stoichiometry:

• We determine the limiting reactant, which, in this case, is the number of caps.
• We can calculate precisely which resource will run out first, dictating the maximum number of bottles that can be capped and filled.

Understanding the stoichiometry in this scenario helps us figure out what constraints are present during production and allows us to predict the leftover resources. This knowledge is essential for efficient production management.

Efficient Bottling Process

The bottling process is an intricate dance between preparation and precision. In the case of our bottling plant example, each component of the process must be precisely calculated and matched to avoid resource wastage.
Firstly, the bottles need to be filled with the exact amount of beverage, which requires an understanding of the volume each bottle can hold. Automation can play a significant role here to ensure every bottle is filled accurately to 355 mL.
After filling, capping is the immediate next step. Each bottle needs a cap to be sealed properly. The caps serve as an integral part of the process, being both a functional and limiting factor.
Key factors for an efficient bottling process include:

• Precise measurement of liquid quantities for consistent bottle filling.
• Coordinated timing between bottle filling and capping machines to match speed and avoid backups.
• Regular checks on equipment to ensure proper functioning and minimal downtime.

A miscalculation in any part of this process can lead to either a surplus or shortage of resources, impacting the overall efficiency.

Understanding Chemical Calculations in Production

Chemical calculations in production involve using mathematical techniques to ensure resources are used effectively. This is especially important in large-scale operations like bottling, where thousands of bottles are processed at a time.
In our exercise, calculations include:

• Converting the beverage from liters to milliliters to match the bottle capacity, ensuring precise measurement.
• Determining the number of bottles that can be filled and capped, requiring the use of division and comparison techniques.
• Calculating the leftover resources to assess future needs or adjustments in supply.

The calculations help identify the limiting reactant - the component that runs out first. This concept is crucial in chemical reactions and manufacturing processes, as it establishes the maximum possible output.
The ability to perform and understand these calculations permits better planning and can lead to cost savings by optimizing resource distribution. By honing these skills, plant operators can ensure smoother transactions and greater productivity in their operations.