Question

select the correct answer.
a company manufactures acoustic guitars. the table shows the number of units that the company sold each month over a nine - month period beginning in january.
month x: 1 2 3 4 5 6 7 8 9
units y: 150 300 380 425 425 480 520 550 575
which equation best models the situation?
a. y = 148.4√(x - 1)-154
b. y = 46.4x + 191.7
c. y = 148.4√(x - 1)+154
d. y=-46.4x + 191.7

Explanation:

Step1: Check linear - form equations

We can test the linear - form equations \(y = mx + b\) (options B and D) first. For a linear equation \(y=mx + b\), the slope \(m=\frac{y_2 - y_1}{x_2 - x_1}\). Let's take two points \((x_1,y_1)=(1,150)\) and \((x_2,y_2)=(2,300)\). The slope \(m=\frac{300 - 150}{2 - 1}=150\).
For option B: \(y = 46.4x+191.7\), when \(x = 1\), \(y=46.4\times1 + 191.7=238.1
eq150\).
For option D: \(y=-46.4x + 191.7\), when \(x = 1\), \(y=-46.4\times1+191.7 = 145.3
eq150\). So options B and D are not correct.

Step2: Check non - linear form equations

Now consider the non - linear form equations (options A and C). Let's test option A: \(y = 148.4\sqrt{x - 1}-154\). When \(x = 1\), \(y=148.4\sqrt{1 - 1}-154=- 154
eq150\).
Let's test option C: \(y = 148.4\sqrt{x - 1}+154\). When \(x = 1\), \(y=148.4\sqrt{1 - 1}+154 = 154\approx150\) (a small difference may be due to rounding in the model). As \(x\) increases, we can substitute other values of \(x\) from the table into \(y = 148.4\sqrt{x - 1}+154\) and see that it fits the data trend better than the other equations.

Answer:

C. \(y = 148.4\sqrt{x - 1}+154\)