Question

A company finds that its sales since the company started in 2000 can be modelled by the function s(t)=(20t^2+800t+300)/(8t^2+10t+100) where s is the total sales, in millions of dollars, and t is the number of years since 2000. a) calculate the years when the sales are 9 million, algebraically.

Answer 1

Solving the quadratic function, the sales are of 9 million in the years of 2000 and 2012.

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 sales(in millions of dollars), in t years after 2000, is modeled by the following function:

$S(t) = \frac{20t^2 + 800t + 300}{8t^2 + 10t + 100}$

The sales are of 9 million when S(t) = 9, hence:

$S(t) = \frac{20t^2 + 800t + 300}{8t^2 + 10t + 100}$

$9 = \frac{20t^2 + 800t + 300}{8t^2 + 10t + 100}$

$72t^2 + 90t + 900 = 20t^2 + 800t + 300$

$52t^2 - 710t + 600 = 0$

Which is a quadratic equation with coefficients a = 52, b = -710, c = 600, hence:

$\Delta = (-710)^2 - 4(52)(600) = 379300$

$x_1 = \frac{710 + \sqrt{379300}}{104} = 12.7$

$x_2 = \frac{710 - \sqrt{379300}}{104} = 0.91$

t is measured in years after 2000, the sales are of 9 million in the years of 2000 and 2012.

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

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