Current Attempt in Progress Harley-Davidson Inc. manufactures motorcycles. During the years following 2018 (the company's 115^th anniversary year), the company's gross income can be approximatec 1^1 by 1.97-0.13t billion dollars per year, where is time in years since January 12018. Assume this rate holds through January 1,2023, and assume a continuous interest rate of 1.9% per year. (a) What was the income of the Harley-Davidson Company in 2018(t=0) What is the projected gross income in 2022 (t=4) The gross income in 2018 was square billion dollars. 1 The projected gross income in 2022 is square billion dollars. 1 (b) What was the present value, on January 1,2018, of Harley -Davidson's gross income for the four years from January 1,2018 to January 1, 2022? Round your answer to two decimal places. square billion dollars i (c) What is the future value, on January 1,2022, of the gross income for the preceding 4 years? square billion dollars i twww.marketwatch.com, accessed December 16, 2020. eTextbook and Media

**(a) Income in 2018 and Projected Income in 2022:**<br /><br />The gross income is given by the function I(t) = 1.97 - 0.13t billion dollars, where t is the time in years since January 1, 2018.<br /><br />* **2018 (t=0):**<br /> I(0) = 1.97 - 0.13 * 0 = 1.97 billion dollars<br /><br />* **2022 (t=4):**<br /> I(4) = 1.97 - 0.13 * 4 = 1.97 - 0.52 = 1.45 billion dollars<br /><br />The gross income in 2018 was $\boxed{1.97}$ billion dollars.<br />The projected gross income in 2022 is $\boxed{1.45}$ billion dollars.<br /><br /><br />**(b) Present Value on January 1, 2018:**<br /><br />The present value of a continuous income stream is given by the integral of the income function multiplied by the discount factor e^(-rt), where r is the continuous interest rate. Here, r = 0.019, and we integrate from t=0 to t=4.<br /><br />PV = ∫₀⁴ (1.97 - 0.13t)e^(-0.019t) dt<br /><br />To solve this integral:<br /><br />1. **Split the integral:** ∫₀⁴ 1.97e^(-0.019t) dt - ∫₀⁴ 0.13te^(-0.019t) dt<br /><br />2. **Solve each part:**<br /> * The first part is a simple exponential integral: (1.97/-0.019)[e^(-0.019t)] from 0 to 4<br /> * The second part requires integration by parts: Let u = t and dv = e^(-0.019t)dt. Then du = dt and v = (-1/0.019)e^(-0.019t). The integral becomes uv - ∫vdu = (-t/0.019)e^(-0.019t) - ∫(-1/0.019)e^(-0.019t) dt. Evaluate this from 0 to 4.<br /><br />After calculating both parts and combining the results, you get:<br /><br />PV ≈ 7.46 billion dollars<br /><br />The present value on January 1, 2018, is approximately $\boxed{7.46}$ billion dollars.<br /><br /><br />**(c) Future Value on January 1, 2022:**<br /><br />The future value (FV) can be calculated by multiplying the present value by e^(rt), where t is the number of years (4) and r is the interest rate (0.019).<br /><br />FV = PV * e^(0.019 * 4)<br />FV ≈ 7.46 * e^0.076<br />FV ≈ 8.05 billion dollars<br /><br />The future value on January 1, 2022, is approximately $\boxed{8.05}$ billion dollars.<br />