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), and $t$ is the time in years.
Given $A = 5000$, $P = 3800$, $r = 0.07$ (since $7\%=0.07$). Substitute these values into the formula:
$$5000 = 3800(1 + 0.07)^t$$

Step2: Simplify the equation

Divide both sides by $3800$:
$$\frac{5000}{3800}=(1.07)^t$$
Simplify $\frac{5000}{3800}\approx1.315789$:
$$1.315789=(1.07)^t$$

Step3: Solve for $t$ using logarithms

Take the natural logarithm of both sides:
$$\ln(1.315789)=\ln((1.07)^t)$$
Using the property of logarithms $\ln(a^b)=b\ln(a)$, we get:
$$\ln(1.315789)=t\ln(1.07)$$
Then solve for $t$:
$$t=\frac{\ln(1.315789)}{\ln(1.07)}$$
Calculate the values: $\ln(1.315789)\approx0.275$, $\ln(1.07)\approx0.0677$. Then $t=\frac{0.275}{0.0677}\approx4.06$ (using a calculator for more precise calculation: $\ln(5000/3800)\approx\ln(1.31578947)\approx0.2753$, $\ln(1.07)\approx0.0676586$, so $t = 0.2753\div0.0676586\approx4.07$)

Answer:

4.07