Question

use a graphing calculator to find the number of years it takes for the investment to grow to $5000. round to the nearest hundredth of a year. a $3800 investment that earns 7% annual interest compounded yearly enter the correct value in the box. show hints years

Explanation:

Step1: Recall compound interest formula

The compound - interest formula is $A = P(1 + r)^{t}$, where $A$ is the final amount, $P$ is the principal amount, $r$ is the annual interest rate (in decimal form), and $t$ is the number of years.
We know that $P=\$3800$, $r = 7\%=0.07$, and $A = \$5000$. We need to solve for $t$.
Substitute the known values into the formula: $5000=3800(1 + 0.07)^{t}$

Step2: Isolate the exponential term

Divide both sides of the equation by $3800$:
$\frac{5000}{3800}=(1.07)^{t}$
Simplify $\frac{5000}{3800}=\frac{25}{19}\approx1.3158$
So we have $1.3158=(1.07)^{t}$

Step3: Take the natural logarithm of both sides

Take the natural log of both sides: $\ln(1.3158)=\ln(1.07^{t})$
Using the property of logarithms $\ln(a^{b})=b\ln(a)$, we get:
$\ln(1.3158)=t\ln(1.07)$

Step4: Solve for t

Divide both sides by $\ln(1.07)$ to solve for $t$:
$t=\frac{\ln(1.3158)}{\ln(1.07)}$
Calculate $\ln(1.3158)\approx0.275$ and $\ln(1.07)\approx0.0677$
Then $t=\frac{0.275}{0.0677}\approx4.06$

Answer:

$4.06$