Question

In the United States, coffee consumption averages \(4.00 \times 10^{8}\) cups per day. If each cup of coffee contains \(160 \mathrm{mg}\) caffeine, how many pounds of caffeine are being consumed each day?

Short answer

Approximately 141 pounds of caffeine are consumed each day in the United States.

Step-by-step solution

- Understand the Units

To solve this problem, first, understand the units involved. You need to find out how many pounds of caffeine are consumed per day, knowing the daily consumption in cups and the caffeine content per cup.

- Calculate Total Caffeine Consumption in Milligrams

Multiply the total number of cups consumed per day by the amount of caffeine per cup to obtain the total caffeine consumption in milligrams. o Total cups consumed per day: \(4.00 \times 10^{8}\)o Caffeine per cup: 160 mgo Total caffeine consumption (in mg):\[4.00 \times 10^{8} \text{ cups} \times 160 \text{ mg/cup} = 6.40 \times 10^{10} \text{ mg}\]

- Convert Milligrams to Pounds

Convert the total caffeine consumption from milligrams to pounds using the conversion factor where 1 pound (lb) equals approximately 453,592.37 milligrams (mg).o Total caffeine in mg: \(6.40 \times 10^{10} \text{ mg}\)o Conversion factor: \(1 \text{ lb} = 453,592.37 \text{ mg}\)o Total caffeine in pounds:\[\frac{6.40 \times 10^{10} \text{ mg}}{453,592.37 \text{ mg/lb}} \ = 141.04 \text{ lbs}\]

Key concepts

dimensional analysis

Dimensional analysis is a technique used to convert one kind of unit to another by using conversion factors. It's really useful in chemistry and other sciences. In this exercise, we use dimensional analysis to convert caffeine consumption from milligrams to pounds.

Conversion factors act as bridges between different units. For example, we know that 1 pound is equal to 453,592.37 milligrams.

To carry out dimensional analysis:

• Start with the amount you want to convert (in this case, caffeine in milligrams).
• Use the conversion factor to cancel out the original unit and get the unit you want.

In this exercise, you start with the total caffeine in milligrams, then multiply it by the appropriate conversion factor to find the caffeine in pounds. It's like solving a puzzle, piece by piece, until you get the complete picture.

mass conversion

Mass conversion involves changing the measurement of a substance's mass from one unit to another. This is critical in solving chemistry problems, like how much caffeine is consumed in pounds instead of milligrams.

Here’s the logical flow for mass conversion used in our exercise:

• First, we calculate the total amount of caffeine consumed in milligrams per day by multiplying the number of cups with the milligrams of caffeine per cup:
  \[ 4.00 \times 10^{8} \text{ cups} \times 160 \text{ mg/cup} = 6.40 \times 10^{10} \text{ mg} \]
• Next, we convert milligrams to pounds. Knowing that 1 pound = 453,592.37 milligrams, we divide the total amount in milligrams by this conversion factor:
  \[ \frac{6.40 \times 10^{10} \text{ mg}}{453,592.37 \text{ mg/lb}} = 141.04 \text{ lbs} \]

This process demonstrates how we can switch between different units of mass by using appropriate conversion factors.

scientific notation

Scientific notation is a method of expressing very large or very small numbers in a compact form. It helps simplify calculations and makes handling extreme values more manageable.

In our exercise, scientific notation is used to express the large number of coffee cups consumed per day and the resulting total caffeine in milligrams.

Here’s a breakdown:

• The number of coffee cups is given as \( 4.00 \times 10^8 \). This means 4 followed by 8 zeros, or 400,000,000 cups.
• When calculating the total caffeine, we use scientific notation again:
  \[ 4.00 \times 10^8 \text{ cups} \times 160 \text{ mg/cup} = 6.40 \times 10^{10} \text{ mg} \]
  That is 64,000,000,000 milligrams.

Finally, we find our answer to the mass conversion problem:

\[ 141.04 \text{ pounds} \] Scientific notation makes it easier to handle these large numbers without making errors in calculations or writing.