Question

1. you invest $1,000 in a certificate of deposit (cd) that earns 5% interest every year. if the function that represents the growth of your money is $f(x) = 1.05x$, in what year will your cd be worth at least $1,300?

year 7
year 4
year 5
year 6

Explanation:

Step1: Set up growth formula

The correct compound growth formula is $A = P(1+r)^t$, where $P=\$1000$, $r=0.05$, $A\geq\$1300$. Substitute values:
$$1000(1.05)^t \geq 1300$$

Step2: Isolate exponential term

Divide both sides by 1000:
$$1.05^t \geq 1.3$$

Step3: Solve for t using logs

Take natural log of both sides, use $\ln(a^b)=b\ln(a)$:
$$t \geq \frac{\ln(1.3)}{\ln(1.05)}$$
Calculate values: $\ln(1.3)\approx0.26236$, $\ln(1.05)\approx0.04879$, so
$$t \geq \frac{0.26236}{0.04879} \approx 5.38$$

Step4: Round up to full year

Since t must be a whole year, round up to 6.

Answer:

Year 6