Human hair grows at a rate of about 6.849×10^−4 cm per hour to 2.329×10^−2 cm per hour. The rate depends on gender, genetics, age, and health. Find the difference between the high end and the low end of the range. Express your answer in scientific notation. Do not round your answer.
The difference between the high end and the low end of the growth rate range for human hair is found by subtracting the lower rate from the higher rate. Let's calculate this in scientific notation.
High end: \(2.329 \times 10^{-2}\) cm per hour
Low end: \(6.849 \times 10^{-4}\) cm per hour
Difference = High end - Low end
Difference = \((2.329 \times 10^{-2}) - (6.849 \times 10^{-4})\)
To subtract these, we need them to have the same exponent. To do this, we can convert \(2.329 \times 10^{-2}\) to an equivalent expression with a power of \(10^{-4}\) by multiplying by \(10^2/10^2\) which is equal to 1 and won't change the value:
\(2.329 \times 10^{-2} = 2.329 \times (10^{-2} \times 10^2/10^2)\)
\(2.329 \times 10^{-2} = 2.329 \times (10^0/10^4)\)
\(2.329 \times 10^{-2} = 2.329 \times 10^{-4} \times 10^2\)
\(2.329 \times 10^{-2} = 232.9 \times 10^{-4}\)
Now we can subtract:
Difference = \((232.9 \times 10^{-4}) - (6.849 \times 10^{-4})\)
Before we perform the subtraction, we align the decimal places:
Difference = \((232.9 - 6.849) \times 10^{-4}\)
Now, we can subtract the decimal numbers:
Difference = \(226.051 \times 10^{-4}\)
To express this in scientific notation, we want only one non-zero digit before the decimal point:
Difference = \(2.26051 \times 10^{2} \times 10^{-4}\)
Difference = \(2.26051 \times 10^{-2}\) (Since \(10^{2}\times 10^{-4} = 10^{-2}\))
The difference between the high end and the low end of the hair growth rate range, without rounding and in scientific notation, is \(2.26051 \times 10^{-2}\) cm per hour.