Solve the problem.
-How long will it take for $8400 to grow to $14.600 at an interest rate of 9.4% if the interest is compounded continuously? Round the number of years to the nearest hundredth.
A) 0.59 yr
B) 0.06 yr
C) 58.81 yr
D) 5.88 yr

Question

Quizplus · Accepted Answer

The continuous compound interest formula is given by $A = Pe^{rt}$, where $A$ is the amount after time $t$, $P$ is the principal, $r$ is the interest rate, and $e$ is Euler's number. We can plug in the given values to get:
$$14,600 = 8,400e^{0.094t}$$
Dividing both sides by $8,400$ and taking the natural logarithm of both sides, we get:
$$\ln \left(\frac{73}{42}ight) = 0.094t$$
Solving for $t$, we get:
$$t \approx 5.88	ext{ years}$$
Rounding to the nearest hundredth, the answer is $\boxed{	ext{(D) }5.88	ext{ yr}}$. We can use the formula for continuous compound interest:

$A = Pe^{rt}$

where A is the final amount, P is the initial amount, r is the interest rate (as a decimal), and t is the time (in years).

We can plug in the given values:

$14600 = 8400e^{0.094t}$

Divide both sides by 8400:

$\frac{14600}{8400} = e^{0.094t}$

Take the natural logarithm of both sides:

$\ln{\frac{14600}{8400}} = 0.094t$

Simplify:

$\ln{\frac{73}{42}} = 0.094t$

$t = \frac{\ln{\frac{73}{42}}}{0.094} \approx 5.88$

Therefore, it will take approximately 5.88 years for the initial amount of $8400 to grow to $14,600 with continuous compound interest at a rate of 9.4%. The closest answer choice is D, which is the correct answer.

Solve the Problem. -How Long Will It Take for $8400 to Grow to to Grow