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Chapter 3: Problem 10

Suppose a country's population increases by a total of \(6 \%\) over a three- year period. What is the continuous growth rate for this country?

Short Answer

Expert verified
The continuous growth rate for this country is approximately 1.98%.

Step by step solution

01

Understand the exponential growth formula

The exponential growth formula is given by: \[P(t) = P_0 e^{rt}\] where: - \(P(t)\) is the population after time t - \(P_0\) is the initial population - \(r\) is the continuous growth rate - \(t\) is the time in years - \(e\) is the base of the natural logarithm, approximately equal to 2.71828. In this problem, we know that the population increased by 6% in 3 years, so we can write: \[P(3) = 1.06P_0\]
02

Plug in the known values

Using the exponential growth formula, we can plug in the known values: \[1.06P_0 = P_0 e^{(3r)}\]
03

Solve for the continuous growth rate r

First, we can eliminate \(P_0\) from both sides of the equation: \[1.06 = e^{(3r)}\] To solve for r, we can take the natural logarithm of both sides: \[\ln{1.06} = \ln{e^{(3r)}}\] Using the properties of natural logarithms, we get: \[\ln{1.06} = 3r\] Now, we can solve for r: \[r = \frac{\ln{1.06}}{3}\] \[r \approx 0.0198\]
04

Convert the continuous growth rate to percentage

To convert the continuous growth rate to percentage, we multiply it by 100: \[Continuous\: Growth\: Rate = 0.0198 * 100 = 1.98 \% \] The continuous growth rate for this country is approximately 1.98%.

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Most popular questions from this chapter

Suppose \(x\) is a positive number. (a) Explain why \(x^{t}=e^{t \ln x}\) for every number \(t\). (b) Explain why $$ \frac{x^{t}-1}{t} \approx \ln x $$ if \(t\) is close to 0

Suppose \(t\) is a small positive number. Estimate the slope of the line containing the points \(\left(4, e^{4}\right)\) and \(\left(4+t, e^{4+t}\right)\)

Show that \(\cosh x \geq 1\) for every real number \(x\).

Show that \(\sinh x \approx x\) if \(x\) is close to 0 [The definition of \(\sinh\) was given before Problem 60 in Section 3.5.]

Show that the range of \(\sinh\) is the set of real numbers.
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