Scientif Notation

Question

The average person takes approximately 30,000 breaths per day.  Kelly decides the average person will take about 900 million breaths in a lifetime, is this reasonable? Mathematically justify your response with scientific notation.

Answer 1

Her answer is reasonable based on the average lifetime of 70 years.

You can calculate the number of breaths a person takes in a lifetime by multiplying the number of breaths per day with the number of days in an average lifetime.  If we assume an average lifetime of 70 years and we use the conversion that there are approximately 400 days in a year, we get...

 

30,000 breaths/day x 400 days /year x 70 years /lifetime = 84 x 10 ^7 breaths /lifetime = 840 x 10^6 breaths/lifetime = 840 billion breaths/lifetime

 

This will give an order of magnitude estimate that is of the same order as the answer that Kelly obtained. 

Answer 2

 9 * 10⁸ / 3 * 10⁴ = 3 * 10⁴   Number of days of lifetime

 

    3 * 10⁴/ 365 = 82 ( approximation for years of life)  

 

  It is a good approximation for life time to breath 9* 10⁸

Answer 3

Express 30,000 in scientific notation

3 x 10^4 (the exponent on the 10 is the number of spaces you move the decimal to the right if positive and to the left if negative)

Express 900 million in scientific notation 900,000,000

9 x 10^8

A one year average breaths in scientific notation can be found by

3 x 10^4 breaths/day x 3.65 x 10^2 days/year

When multiplying scientific notation, multiply the numbers and add the exponents

(3 x 3.65) x 10^(4+2) = 10.95 x 10^6 = 1.095 x 10^7 breaths/year

Taking the estimate of (9 x 10^8 breaths/lifetime)/(1.095 x 10^7 breaths/year) = 8.2 x 10^(8-7) = 8.2 x 10^1 = 82 years seems reasonable for a lifetime.

OR

Change to .1095 x 10^8 breaths/year to compare to the lifetime breaths of 9 x 10^8 and we see that we would need to multiply the breaths/year by a number greater than 10 and less than 100 to end up with the 900 million so it makes sense.