Question

the data points show the amount of money y (in dollars) in an account after a time x (in years). each figure has the same data points. however, each figure has a different curve fitting the data. the equation for each curve is also shown. answer the questions that follow. (a) which curve fits the data best? figure 1 figure 2 figure 3 (b) use the equation of the best fitting curve from part (a) to predict the amount of money in the account after 37 years. round your answer to the nearest

Explanation:

Step1: Observe the data - point distribution

Visually, the points seem to follow an exponential - like growth pattern. The linear function in Figure 1 ($y = 40x+400$) does not capture the upward - curving nature of the data well. The quadratic function in Figure 3 ($y = 3x^{2}-75x + 900$) has a vertex and does not fit the overall upward trend. The exponential function in Figure 2 ($y=609(1.05)^{x}$) closely follows the data points.

Step2: Predict using the best - fitting curve

We use the equation $y = 609(1.05)^{x}$ from Figure 2. Substitute $x = 37$ into the equation:
\[y=609\times(1.05)^{37}\]
First, calculate $(1.05)^{37}$. Using the formula $a^{n}=e^{n\ln(a)}$, we have $\ln(1.05)\approx0.04879$ and $n = 37$, so $n\ln(1.05)=37\times0.04879 = 1.70523$. Then $(1.05)^{37}=e^{1.70523}\approx5.5078$.
Multiply by 609: $y=609\times5.5078\approx3354.25$.

Answer:

(a) Figure 2
(b) $\$3354$