Question

The following formula models the number of automobiles, A,in thousands, that Washington residents purchased

x years after 1980.
A=2x²+24x+300

Write an equation that you can use to determine the first year in which residents purchased approximately 566 thousand cars.

Answer 1

To determine the first year in which residents purchased approximately 566 thousand cars, we can solve the given equation. By substituting the given value for A, we can use the quadratic formula to solve for x. The first year in which residents purchased approximately 566 thousand cars is 1987.

Step-by-step explanation:

To determine the first year in which residents purchased approximately 566 thousand cars, we need to solve the equation A = 2x² + 24x + 300 for x.

Let's substitute A with 566 and solve the equation:

1. 566 = 2x² + 24x + 300
2. 2x² + 24x + 300 - 566 = 0
3. 2x² + 24x - 266 = 0
4. Divide the equation by 2: x² + 12x - 133 = 0
5. Using the quadratic formula: x = (-b ± sqrt(b² - 4ac)) / (2a)
6. Substituting the values into the formula, we get: x = (-12 ± sqrt(12² - 4(1)(-133))) / (2(1))
7. Simplifying further, we have: x = (-12 ± sqrt(144 + 532)) / 2
8. Continuing the calculations: x = (-12 ± sqrt(676)) / 2
9. Finally, we have two possible solutions: x = (-12 + 26) / 2 = 7 or x = (-12 - 26) / 2 = -19.

Since we are looking for the years after 1980, we discard the negative solution.

Therefore, the first year in which residents purchased approximately 566 thousand cars is 1987.