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Chapter 4: Problem 15

An average of \(1.5 \cdot 10^{4}\) Coca-Cola beverages were consumed every second worldwide in \(2005 .\) There are \(8.64 \cdot 10^{4}\) seconds in a day. What was the daily consumption of CocaCola in 2005 ?

Short Answer

Expert verified
The daily consumption of Coca-Cola in 2005 was \(1.296 \cdot 10^{9}\) beverages.

Step by step solution

01

Understand the Given Values

Identify the given values in the problem: the average number of Coca-Cola beverages consumed per second and the number of seconds in a day. Specifically, the average consumption per second is given as \(1.5 \cdot 10^{4}\) beverages, and there are \(8.64 \cdot 10^{4}\) seconds in a day.
02

Set up the Multiplication

To find the total daily consumption, multiply the average number of beverages consumed per second by the number of seconds in a day. The expression to calculate this is \( (1.5 \cdot 10^{4}) \times (8.64 \cdot 10^{4}) \).
03

Perform the Multiplication

Break down the multiplication into two parts: multiplying the decimal parts and adding the exponents separately. Calculate \( 1.5 \times 8.64 = 12.96\), and then add the exponents \( 10^{4} \times 10^{4} = 10^{8} \). The result of the multiplication is \( 12.96 \cdot 10^{8} \).
04

Express the Final Answer in Scientific Notation

Convert \( 12.96 \cdot 10^{8} \) into proper scientific notation. This provides \(1.296 \cdot 10^{9}\) as the final answer, since moving the decimal point one place to the left reduces the 12.96 to 1.296 and increases the exponent by 1.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Daily consumption calculation
To calculate daily consumption of Coca-Cola beverages, begin by identifying the relevant values. We know the average number of drinks consumed per second worldwide in 2005 is given by the scientific notation: \(1.5 \times 10^{4}\) beverages per second.
Next, we have the number of seconds in a day which is: \(8.64 \times 10^{4}\) seconds.
The total daily consumption can be calculated by multiplying these two values together.
Mathematically, this is: \((1.5 \times 10^{4}) \times (8.64 \times 10^{4})\)
This multiplication will give the total number of Coca-Cola beverages consumed in a day.
Multiplication of exponents
When multiplying numbers in scientific notation, handle the decimal parts and the exponential parts separately. For our example, first, multiply the decimal values: \(1.5 \times 8.64 = 12.96\).
Then, deal with the exponents: when you multiply exponents with the same base (10 in this case), you simply add them together. So, \(10^{4} \times 10^{4} = 10^{8}\).
Combining these results, you get: \(12.96 \times 10^{8}\).
This process keeps the problem manageable and makes it simpler to understand the multiplication of large numbers.
Conversion to scientific notation
To convert a number into proper scientific notation, you want a decimal number between 1 and 10 multiplied by an appropriate power of 10. In our case, we have the number \(12.96 \times 10^{8}\).
We need to convert \(12.96\) to a number within 1 and 10 by moving the decimal point one place to the left, which gives us \(1.296\).
When the decimal point is moved left, the exponent increases by 1. Therefore, \(12.96 \times 10^{8}\) becomes \(1.296 \times 10^{9}\).
This step is crucial for expressing large numbers in a more compact and readable form.
Mathematical problem-solving
Approaching problems systematically helps break down complex calculations. Let's review the process:
  • Identify the given values.
  • Set up the necessary multiplication.
  • Perform multiplication in parts - decimals and exponents separately.
  • Convert the result to proper scientific notation.
Understanding each part of the problem ensures clarity and accuracy. Follow these steps to tackle various mathematical problems effectively. This structured methodology will reinforce your problem-solving skills and make challenging problems more approachable.

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Most popular questions from this chapter

An ant is roughly \(10^{-3}\) meter in length and the average human roughly one meter. How many times longer is a human than an ant?

The distance that light travels in 1 year (a light year) is \(5.88 \cdot 10^{12}\) miles. If a star is \(2.4 \cdot 10^{8}\) light years from Earth, what is this distance in miles?

The National Institutes of Health guidelines suggest that adults over 20 should have a body mass index, or BMI, under \(25 .\) This index is created according to the formula $$ \mathrm{BMI}=\frac{\text { weight in kilograms }}{(\text { height in meters })^{2}} $$ a. Given that 1 kilogram \(=2.2\) pounds, and 1 meter \(=39.37\) inches, calculate the body mass index of President George W. Bush, who is 6 feet tall and in 2003 weighed 194 pounds. According to the guidelines, how would you describe his weight? b. Most Americans don't use the metric system. So in order to make the BMI easier to use, convert the formula to an equivalent one using weight in pounds and height in inches. Check your new formula by using Bush's weight and height, and confirm that you get the same BMI. c. The following excerpt from the article "America Fattens Up" (The New York Times, October 20,1996 ) describes a very complicated process for determining your BMI: To estimate your body mass index you first need to convert your weight into kilograms by multiplying your weight in pounds by 0.45. Next, find your height in inches. Multiply this number by 0.254 to get meters. Multiply that number by itself and then divide the result into your weight in kilograms. Too complicated? Internet users can get an exact calculation at http://141.106.68.17/bsa.acgl. Can you do a better job of describing the process? d. A letter to the editor from Brent Kigner, of Oneonta, N.Y., in response to the New York Times article says: Math intimidates partly because it is often made unnecessarily daunting. Your article "American Fattens \(U p "\) comvolutes the procedure for calculating the Body Mass Index so much that you suggest readers retreat to the Internet. In fact, the formula is simple: Multiply your weight in pounds by 703 , then divide by the square of your height in inches. If the result is above \(25,\) you weigh too much. Is Brent Kigner right?

For each equation determine the value of \(x\) that makes it true. a. \(10^{x}=0.000001\) b. \(10^{x}=\frac{1}{1,000,000}\) c. \(\frac{1}{10^{x}}=0.0001\) d. \(10^{-x}=100,000\)

A TV signal traveling at the speed of light takes about \(8 \cdot 10^{-5}\) second to travel 15 miles. How long would it take the signal to travel a distance of 3000 miles?
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