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Chapter 3: Problem 12

Suppose the number of cell phones in the world increases by a total of \(150 \%\) over a five-year period. What is the continuous growth rate for the number of cell phones in the world?

Short Answer

Expert verified
The continuous growth rate for the number of cell phones in the world is approximately 18.326%. We can find it by using the formula \( r = \frac{\ln(\frac{A}{P})}{t} \), where A is the final amount (2.5 times the initial amount) and t is the time (5 years).

Step by step solution

01

Determine the final amount given an increase of 150%

Since the amount of cell phones increases by 150% in 5 years, that means the final number of cell phones, A, is 1 + the 150% increase: \( A = 1 + 1.50 = 2.50 \).
02

Rewrite the continuous compound interest formula to solve for the interest rate r

The formula for continuous compound interest is: \( A=P * e^{rt} \). We want to find the interest rate, 'r,' so we will rewrite the formula: \( r = \frac{\ln(\frac{A}{P})}{t} \).
03

Input the given values to find the continuous growth rate r

Now, we will input the values of A, P, and t that we either know or determined previously: \( r = \frac{\ln(\frac{2.50}{1})}{5} \) Calculate the natural logarithm value: \( r = \frac{\ln(2.50)}{5} \) Now, divide the natural logarithm value by 5 to find the growth rate: \( r \approx \frac{0.9163}{5} = 0.18326 \)
04

Convert the growth rate to a percentage

Finally, we convert the growth rate to a percentage by multiplying it by 100: \( r \approx 0.18326 * 100 = 18.326 \% \) Therefore, the continuous growth rate for the number of cell phones in the world is approximately 18.326%.

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