Question

jenny has $250 in a savings account that earns 3% compound interest annually. after how many years will the account balance be more than $750? express this situation as an exponential inequality and solve it. let t represent the time in years. round your answer to the nearest hundredth. use the keypad to enter the answers in the boxes. the exponential inequality that represents this situation is. the account balance will be more than $750 after years.

Explanation:

Step1: Recall compound interest formula

The formula for annual compound interest is $A = P(1+r)^t$, where $P=\$250$, $r=0.03$, and $A>\$750$.

Step2: Write the exponential inequality

Substitute values into the formula:
$250(1+0.03)^t > 750$

Step3: Simplify the inequality

Divide both sides by 250:
$(1.03)^t > 3$

Step4: Solve for t using logarithms

Take natural log of both sides:
$\ln(1.03^t) > \ln(3)$
Use log power rule $\ln(a^b)=b\ln(a)$:
$t\ln(1.03) > \ln(3)$
Isolate t:
$t > \frac{\ln(3)}{\ln(1.03)}$

Step5: Calculate the value

Compute the right-hand side:
$t > \frac{1.098612}{0.029559} \approx 37.17$

Answer:

The exponential inequality that represents this situation is $250(1.03)^t > 750$.
The account balance will be more than $\$750$ after $37.17$ years.