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) We can fill and cap 115,211 bottles. (b) There will be 6,304 bottles, 7,289 caps, and approximately 221.97 L of beverage left over. (c) The component that limits the production is the beverage volume.

Step-by-step solution

Identify the limiting factor for production

First, let's find the maximum number of bottles we can fill with the provided beverage volume. To do this, we will divide the total volume of the beverage (40,875 L) by the capacity of each bottle (355 mL). Make sure to convert either liters to milliliters or milliliters to liters before dividing. We will convert liters to milliliters. \[1 \mathrm{L} = 1000 \mathrm{mL}\] So, 40,875 L equals 40,875 * 1000 mL. Now, we can find the number of bottles that can be filled: \[\frac{40,875 * 1000 \mathrm{mL}}{355 \mathrm{~mL}} = 115211.267606\] As we cannot fill a fraction of a bottle, we will round down to the nearest whole number to get 115,211 bottles. Now, we need to compare the number of bottles, caps, and the capacity to be filled based on the beverage quantity to find the limiting factor. In this case, it will be the minimum of these quantities.

Calculate the number of bottles that can be filled and capped

Let's find the minimum among bottles, caps, and the number of bottles we can fill with the beverage so we can determine how many bottles can be properly filled and capped. \[\min(121,515; 122,500; 115,211) = 115,211\] So, we can fill and cap 115,211 bottles. (a) Answer: We can fill and cap 115,211 bottles.

Calculate the amount left of each component

Now, let's find the remaining quantities of each component after we have filled and capped 115,211 bottles. To do that, we will subtract the number of filled and capped bottles from the initial quantities of bottles, caps, and the volume of beverage. - Remaining bottles: \(121,515 - 115,211 = 6,304\) - Remaining caps: \(122,500 - 115,211 = 7,289\) - Remaining beverage: \(40,875 * 1000 - 115,211 * 355 = 221,973.239\) mL or \(221.973239\) L (b) Answer: There will be 6,304 bottles, 7,289 caps, and approximately 221.97 L of beverage left over.

Identify the component that limits production

We have already identified the limiting factor while finding the number of bottles that can be filled and capped. In our case, the beverage volume limited the production, as we found out that the maximum number of bottles that can be filled with the available beverage quantity is 115,211. Both the available bottles and caps were sufficient to cap more than 115,211 bottles, but the lesser amount of beverage limited the production. (c) Answer: The component that limits the production is the beverage volume.