the data points show the amount of money y (in dollars) in an account after a time x (in years). each figure…

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

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

Answer

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$