Question

if $10,000 is invested at an interest rate of 3.5% each year, what is the value of the investment in 7 years? write an exponential function and solve.
a. $y = 10000(1.035)^{-7}$ $7850.01$
b. $y = 10000(1.035)^{7}$ $12,722.70$
c. $y = 10000(7)^{1.035}$ $74,933.58$
d. $y = 10000(715)^{7}$ $64,239.30$
please select the best answer from the choices provided

a
b
c
d

Explanation:

Step1: Recall compound interest formula

The formula for compound interest (exponential growth) is $y = P(1 + r)^t$, where $P$ is principal, $r$ is rate, $t$ is time. Here, $P = 10000$, $r = 0.035$ (3.5% as decimal), $t = 7$. So the function should be $y = 10000(1 + 0.035)^t = 10000(1.035)^t$.

Step2: Substitute $t = 7$

Calculate $y = 10000(1.035)^7$. Compute $(1.035)^7$: $1.035^7 \approx 1.272279$. Then $y \approx 10000 \times 1.272279 \approx 12722.79$? Wait, no, wait the options: Wait option B is $y = 10000(1.035)^t$, $t = 7$? Wait no, wait the options: Wait option B's value is $112,722.79$? Wait no, maybe I miscalculated. Wait $1.035^7$: let's recalculate. $1.035^1 = 1.035$; $1.035^2 = 1.035×1.035 = 1.071225$; $1.035^3 = 1.071225×1.035 ≈ 1.108717875$; $1.035^4 ≈ 1.108717875×1.035 ≈ 1.1475230006$; $1.035^5 ≈ 1.1475230006×1.035 ≈ 1.1876863056$; $1.035^6 ≈ 1.1876863056×1.035 ≈ 1.2292553263$; $1.035^7 ≈ 1.2292553263×1.035 ≈ 1.2722792627$. Then $10000×1.2722792627 ≈ 12722.79$? But option B says $112,722.79$? Wait maybe the time is 70? No, the question says 7 years. Wait maybe the options have a typo, but the function is $y = 10000(1.035)^t$. So the correct function is $y = 10000(1.035)^t$, and when $t = 7$, it's approximately $12722.79$, but option B is $y = 10000(1.035)^t$, $112,722.79$? Wait no, maybe I misread the time. Wait the question says "in 7 years"? Wait no, maybe 70? No. Wait the options: option B is $y = 10000(1.035)^t$, $112,722.79$? Wait $1.035^{40} ≈ 4.0276$, $10000×4 = 40000$, no. Wait maybe the rate is 35%? No, the question says 3.5%. Wait the options: option B's function is $y = 10000(1.035)^t$, and the value is $112,722.79$? Wait no, $1.035^{40}$ is about 4, $1.035^{50}$ is about 5.5, $1.035^{60}$ is about 7.6. Wait maybe the time is 70? No. Wait the original problem: maybe a typo, but the correct function is $y = P(1 + r)^t$, so the function is $y = 10000(1.035)^t$, which is option B's function. So the answer is B.

Answer:

B. $y = 10000(1.035)^t$, $112,722.79$