Question

type the correct answer in the box. round your answer to the nearest whole number.
an investment earning interest at the rate of 10%, compounded continuously, will double in t years. find t.
use the formula $p_t = p_0e^{rt}$, where $p_t$ is the amount after t years, $p_0$ is the initial amount, r is the rate of interest, and t is the time.
t = \boxed{} years.

Explanation:

Step1: Set up the equation

Since the investment doubles, \( P_t = 2P_0 \) and \( r = 0.10 \) (10% as a decimal). Substitute into the formula \( P_t = P_0e^{rt} \):
\( 2P_0 = P_0e^{0.10t} \)
Divide both sides by \( P_0 \) (assuming \( P_0
eq 0 \)):
\( 2 = e^{0.10t} \)

Step2: Take natural logarithm

Take the natural log of both sides:
\( \ln(2) = \ln(e^{0.10t}) \)
Using the property \( \ln(e^x) = x \), we get:
\( \ln(2) = 0.10t \)

Step3: Solve for t

Divide both sides by \( 0.10 \):
\( t = \frac{\ln(2)}{0.10} \)
Calculate \( \ln(2) \approx 0.6931 \), so:
\( t \approx \frac{0.6931}{0.10} = 6.931 \)
Round to the nearest whole number: \( t \approx 7 \)

Answer:

7