Question

apply what you have learned!
as with other types of equations, you can solve exponential functions for any variable. in this section, we will solve for the exact initial value but only estimate the exponent since solving for exponents is beyond what we have learned so far.

1. sari makes an investment with the goal of having $20,000 after 30 years. if the growth factor is 1.12, how much money will she need to invest to reach her goal?
2. charlotte has $12,000 to invest and would like this investment to grow to $18,000. if the growth factor is 1.09, how many years will it take to reach her goal? use guess and check to find an approximate answer.

Explanation:

Step1: Recall compound growth formula

The formula for exponential growth is $A = P(r)^t$, where $A$ is the final amount, $P$ is the initial principal, $r$ is the growth factor, and $t$ is time. Rearrange to solve for $P$: $P = \frac{A}{r^t}$

Step2: Substitute values for Sari's investment

$A = 20000$, $r = 1.12$, $t = 30$
$P = \frac{20000}{(1.12)^{30}}$
Calculate $(1.12)^{30} \approx 29.9599$
$P \approx \frac{20000}{29.9599} \approx 667.55$

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Step1: Use growth formula for Charlotte's goal

Start with $A = P(r)^t$, substitute $A=18000$, $P=12000$, $r=1.09$:
$18000 = 12000(1.09)^t$
Simplify: $1.5 = (1.09)^t$

Step2: Guess and check for t

• Try $t=4$: $(1.09)^4 \approx 1.4116 < 1.5$
• Try $t=5$: $(1.09)^5 \approx 1.5386 > 1.5$

The value 1.5 is between $t=4$ and $t=5$. Test $t=4.8$:
$(1.09)^{4.8} = (1.09)^4 \times (1.09)^{0.8} \approx 1.4116 \times 1.0703 \approx 1.510$
Test $t=4.7$:
$(1.09)^{4.7} = (1.09)^4 \times (1.09)^{0.7} \approx 1.4116 \times 1.0626 \approx 1.499 \approx 1.5$

Answer:

1. Sari needs to invest approximately $\$667.55$.
2. It will take approximately 4.7 years for Charlotte's investment to reach her goal.