Question

The following formula models the number of automobiles, A,in thousands, that Washington residents purchased
x years after 1980.
A=2x2+24x+300

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


B) Now solve your equation and specify the first year (for example, enter 2019 and not 39) in which residents purchased approximately 566 thousand cars.

Answer 1

Considering the given quadratic equation for the number of automobiles, we have that:

A) The equation to be solved is: x² + 12x - 133 = 0

B) The first year was of 1987.

What is a quadratic function?

A quadratic function is given according to the following rule:

$y = ax^2 + bx + c$

The solutions are:

• $x_1 = \frac{-b + \sqrt{\Delta}}{2a}$

• $x_2 = \frac{-b - \sqrt{\Delta}}{2a}$

In which:

$\Delta = b^2 - 4ac$

The number of automobiles, in x years after 1980, is modeled by:

A(x) = 2x² + 24x + 300.

The amount is of 566,000 when A(x) = 566, hence the equation is:

2x² + 24x + 300 = 566

2x² + 24x - 266 = 0

x² + 12x - 133 = 0

The coefficients are a = 1, b = 12, c = -133, hence:

• $\Delta = 12^2 - 4(1)(-133) = 676$

• $x_1 = \frac{-12 + \sqrt{676}}{2} = 7$

• $x_2 = \frac{-12 - \sqrt{676}}{2} = -19$

Time is a positive measure, hence we take the solution of 7, 1980 + 7 = 1987, which is the year.

More can be learned about quadratic functions at https://brainly.com/question/24737967

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