Question

A bottling plant has 121,515 bottles with a capacity of \(355 \mathrm{~mL}, 122,500\) caps, and 40,875 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) The bottling plant can fill and cap 115,070 bottles. (b) After production: 6,445 leftover bottles, 7,430 leftover caps, and 2,286.975 L of leftover beverage. (c) The limiting component is the available beverage volume.

Step-by-step solution

Calculate the number of bottles that can be filled

To find out how many bottles the plant can fill, we need to divide the total volume of the beverage (in liters) by the volume of each bottle (in liters). First, convert the volume of each bottle from milliliters to liters: \( 355~mL = 0.355~L \) Next, divide the total volume of the beverage by the volume of a single bottle. \[ \frac{40,875}{0.355} \approx 115,070 \] Our calculation shows that the plant can fill approximately 115,070 bottles with the available beverage.

Calculate the number of bottles that can be capped

We already have the number of caps available, which is 122,500. Therefore, the plant can cap 122,500 bottles.

Determine which component limits the production

In order to find the limiting component, we need to compare the number of bottles that can be filled, the number of bottles that can be capped, and the total number of available bottles. \( 115,070 \leq 121,515 \) \( 115,070 \leq 122,500 \) Since 115,070 is the smallest value among the available resources, it means the available beverage volume limits the production. Hence, only 115,070 bottles can be filled and capped.

Answer (a)

(a) The bottling plant can fill and cap 115,070 bottles.

Calculate the amount of leftover resources

To find the leftover resources, we have to calculate the difference between the initial amount and the used amount. For bottles: \( 121,515 - 115,070 = 6,445 \) bottles For caps: \( 122,500 - 115,070 = 7,430 \) caps To calculate the amount of remaining beverage, we need to multiply the number of bottles that were not filled (6,445 bottles) by the volume of each bottle (0.355 L). \( 6,445 * 0.355 = 2,286.975~L \)

Answer (b)

(b) After production: - There are 6,445 leftover bottles; - There are 7,430 leftover caps; - There are 2,286.975 L of leftover beverage.

Answer (c)

(c) The component that limits the production is the available beverage volume.

Key concepts

Stoichiometry

Stoichiometry is a branch of chemistry that deals with the quantitative relationships between the reactants and products in chemical reactions. In the context of a bottling plant, stoichiometry can be applied to determine the exact amounts of materials needed for the bottling process. This ensures that resources are utilized efficiently without any wastage.

To put stoichiometry into practice in such a setting, we start by understanding the volume of product that each bottle can contain, often referred to as its capacity. In the case of the exercise, each bottle can hold 355 mL of a beverage. By calculating how many such bottles can be filled with the available volume of beverage, stoichiometry helps us to ascertain that not a single drop of beverage is wasted.

It's vital for students to grasp that stoichiometry isn't just about mixing chemicals in a lab; it's about the careful planning and scaling of production in real-world applications like our bottling plant example. This concept highlights the importance of having a balanced ratio between different parts of a process to achieve efficiency and prevent resource wastage.

Bottling Plant Operations

Bottling plant operations involve several steps: cleaning the bottles, filling them with the product, capping them, and preparing them for distribution. In each stage, managing resources efficiently is key to successful production. The exercise above exemplifies one crucial part of the operation: determining how many bottles can be filled and capped with the available resources.

For successful execution of the bottling process, it’s not enough to have vast quantities of bottles, caps, and beverage—what's equally important is the balance between these components. From the exercise, we’ve learned that having an excess of one resource (like caps) does not translate to increased production unless the other resources (bottles and beverage) are available in corresponding amounts.

Bottling Efficiency and Limitations

In practical terms, bottling plant operations are all about efficiency. That's figuring out how to best utilize the materials at hand and identifying the limiting reagent (or component)—the one that's in short supply and therefore determines the extent of production. In our example, it was the beverage volume that limited the number of bottles that could be filled.

Unit Conversion

Unit conversion is fundamental in many scientific and engineering problems, as it allows for the comparison and calculation of different quantities. In the bottling plant problem, performing unit conversion is essential since it enables us to work with consistent units for volumes (the move from milliliters to liters).

Students should note that a firm grasp of unit conversion can mean the difference between accurate calculations and costly mistakes in both academic and real-world scenarios. When dealing with unit conversion, remember to use established conversion factors — in this case, 1,000 milliliters make up 1 liter.

Understanding how to convert between units ensures that any calculations carried out in the planning or execution phases of an operation will be accurate. This skill is invaluable, especially in a production environment where precision is crucial for maximizing output and minimizing leftover resources.