1. **State the problem:** We are given data of total personal income in the US from 1960 to 2024 and asked to model it with an exponential function of the form $$y = a(1+r)^x$$ where $x$ is years past 1960 and $y$ is income in billions. 2. **Write the exponential model:** From the problem, the model is given as $$y = 411.500(1.074)^x$$ where $a=411.500$ is the income in 1960 and $r=0.074$ is the growth rate. 3. **Check the model for 2018:** For 2018, $x=2018-1960=58$. Calculate the model's prediction: $$y = 411.500(1.074)^{58}$$ 4. **Calculate the value:** $$y = 411.500 \times (1.074)^{58}$$ Using a calculator, $(1.074)^{58} \approx 46.45$, so $$y \approx 411.500 \times 46.45 = 19,120.175$$ 5. **Compare with actual 2018 income:** Actual income is 19,129.6 billion, model predicts 19,120.175 billion, so the model slightly underestimates the income. 6. **Find the year income reaches 68 trillion:** We want to find $x$ such that $$y = 68,000,000$$ (since 68 trillion = 68,000,000 billion) Set up the equation: $$68,000,000 = 411.500(1.074)^x$$ Divide both sides by 411.500: $$\frac{68,000,000}{411.500} = (1.074)^x$$ Simplify: $$165,230.7 = (1.074)^x$$ Take natural logarithm on both sides: $$\ln(165,230.7) = x \ln(1.074)$$ Calculate: $$x = \frac{\ln(165,230.7)}{\ln(1.074)}$$ Using a calculator: $$\ln(165,230.7) \approx 12.015, \quad \ln(1.074) \approx 0.0713$$ So: $$x \approx \frac{12.015}{0.0713} = 168.5$$ 7. **Convert $x$ to year:** $$\text{Year} = 1960 + 168.5 = 2128.5$$ So the model predicts total personal income will reach 68 trillion around mid-year 2128. **Final answers:** - (a) $$y = 411.500(1.074)^x$$ - (b) The model underestimates the 2018 income. - (c) The income reaches 68 trillion around the year 2128.