Final answer:
To solve the quadratic equation t² + 10t - 2000 = 0, the quadratic formula is used. The regression line for distance over time since the year 2000 is y = (2.0 km/min) × t. In economics, substituting tax rate T with 0.3Y simplifies the equation to solve for national income Y.
Step-by-step explanation:
Solving the Quadratic Equation
To solve the quadratic equation t² + 10t - 2000 = 0, we can use the quadratic formula, which is t = (-b ± √(b² - 4ac)) / (2a). Here, a, b, and c represent the coefficients of the terms in the quadratic equation, which are 1, 10, and -2000 respectively.
The quadratic formula will yield two solutions for t, which are the points in time since the year 2000 in this context.
As for the equation of the regression line y = (2.0 km/min) × t, here t denotes the time in minutes with t=0 corresponding to the year 2000. This line reflects a constant increase in distance over time, indicating a steady speed.
In the context of the economic model, when we substitute the tax rate T with 0.3Y, we revise the equation to Y = 140 + 0.9(Y - 0.3Y) + 400 + 800 + 600 - 0.15Y, which simplifies to Y = 1940 + 0.57Y. This step eliminates the second variable, making it easier to solve for Y, the national income.