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Chapter 3: Problem 11

Suppose the amount of the world's computer hard disk storage increases by a total of \(200 \%\) over a fouryear period. What is the continuous growth rate for the amount of the world's hard disk storage?

Short Answer

Expert verified
The continuous growth rate for the world's hard disk storage is approximately \(27.47\%\) per year.

Step by step solution

01

Set Up the Equation

Since the world's computer hard disk storage increases by \(200\%\) in four years, we know that the final amount (A) is \(200\%\) more than the initial amount (P), which means it's \(300\%\) of the initial amount (or 3 times the initial amount). So, we can set up the equation: \(A = P \cdot e^{rt}\) where A = 3P t = 4 years We need to solve for the growth rate, r.
02

Insert A and P into the Equation

Replace A with 3P in the equation: \(3P = P \cdot e^{4r}\)
03

Solve for r

To solve for r, first, we need to divide both sides of the equation by P: \(\frac{3P}{P} = \frac{P \cdot e^{4r}}{P}\) This simplifies to: \(3 = e^{4r}\) Now, take the natural logarithm (ln) of both sides to isolate r: \(ln(3) = ln(e^{4r})\) Since the natural logarithm of e raised to a certain power equals the power itself, we have: \(ln(3) = 4r\) Finally, divide both sides by 4: \(r = \frac{ln(3)}{4}\)
04

Calculate the Continuous Growth Rate

Now, we can plug in the value of ln(3) and divide by 4 to get the continuous growth rate, r: \(r = \frac{ln(3)}{4} \approx 0.2747\) So, the continuous growth rate for the world's hard disk storage is approximately \(27.47\%\) per year.

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