Question

a computer repair technician records the time spent on various projects and the amount of money earned each week. the model \\( \hat{y} = 45.5x + 115 \\) explains the relationship between \\( x \\), the number of hours worked, and \\( \hat{y} \\), the predicted amount of money earned in dollars. what is the \\( y \\)-intercept, and what does it represent in the context?

the \\( y \\)-intercept is 115. when no time is spent working, the technician is predicted to earn \\$115.

the \\( y \\)-intercept is 45.5. when no time is spent working, the technician is predicted to earn \\$45.50.

the \\( y \\)-intercept is 115. for every additional hour worked, the technician is predicted to earn an increase of \\$115.

the \\( y \\)-intercept is 45.5. for every additional hour worked, the technician is predicted to earn an increase of \\$45.50.

Explanation:

The equation of a line is in the form \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y\)-intercept. In the given model \(\hat{y}=45.5x + 115\), comparing it with \(y = mx + b\), we can identify the \(y\)-intercept. The \(y\)-intercept (\(b\)) is the value of \(y\) when \(x = 0\). In the context of the problem, \(x\) represents the number of hours worked. When \(x = 0\) (no time is spent working), \(\hat{y}=45.5\times0 + 115=115\).

Answer:

The \(y\)-intercept is \(115\). When no time is spent working, the technician is predicted to earn \(\$115\).