Question

select the correct answer from each drop-down menu.

the finance department at a regional toy company has been tracking the income and costs of a new line of dolls. they have determined that the income and costs can be modeled by the equations below, where \\(x\\) is the number of dolls sold, in hundreds, and \\(y\\) is the total dollar amount, in thousands.

\\
\

$$\begin{aligned} \\text{income:} \\quad y &= -0.4x^2 + 3x + 45 \\\\ \\text{costs:} \\quad y &= 1.5x + 20 \\end{aligned}$$

\\

consider the system of equations that can be used to determine the number of dolls for which the company will break-even.

how many total possible solutions of the form \\((x,y)\\) are there for this situation? choose an answer
of any possible solutions of the form \\((x,y)\\), how many are viable for this situation? choose an answer

Explanation:

Equate the income and cost equations

$$ -0.4x^2 + 3x + 45 = 1.5x + 20 $$

$$ -0.4x^2 + 1.5x + 25 = 0 $$

$$ 4x^2 - 15x - 250 = 0 $$

Solve the quadratic equation for x

$$ x = \frac{-(-15) \pm \sqrt{(-15)^2 - 4(4)(-250)}}{2(4)} $$

$$ x = \frac{15 \pm \sqrt{225 + 4000}}{8} = \frac{15 \pm \sqrt{4225}}{8} $$

$$ x = \frac{15 \pm 65}{8} $$

$$ x_1 = \frac{80}{8} = 10,\quad x_2 = \frac{-50}{8} = -6.25 $$

Determine the number of total and viable solutions

$$ \text{Total mathematical solutions } (x, y): 2 $$

$$ \text{Since } x \ge 0 \text{ is required for physical quantities of dolls sold:} $$

$$ \text{Viable solutions: } 1 \quad (\text{only } x = 10) $$

Answer:

Question 1

How many total possible solutions of the form (x,y) are there for this situation? <blank>2</blank>

Question 2

Of any possible solutions of the form (x,y), how many are viable for this situation? <blank>1</blank>