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?

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 = \(\frac{48,775 \times 1000 \mathrm{~mL}}{355 \mathrm{~mL}}\)

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

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)

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)\(\times 355 \mathrm{~mL}\)

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.