Question

dr. shah visits patients at their homes. the equation y = 150x + 50 represents the amount she charges, where y represents the total cost and x represents the number of hours she spends at the home. when the equation is graphed, what does the y-intercept represent?
(options: the total for the entire visit, the hourly rate only, the rate for the first 20 minutes, the setup amount after the hourly rate)

Explanation:

Step1: Recall Slope-Intercept Form

The equation is in slope - intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y\)-intercept. Here, \(y = 150x+50\), so \(m = 150\) and \(b = 50\).

Step2: Analyze the Variables

\(x\) is the number of hours spent at the home. When \(x = 0\) (no hours spent at the home), we calculate \(y\): \(y=150(0)+50 = 50\). This \(y\)-intercept (\(b = 50\)) represents the cost when \(x = 0\) (no hours of home visit), which is a fixed cost or the cost before considering the hours spent (like a base charge). Among the options, "the acting - incident after the hourly rate" doesn't make sense. "the total for the entire visit" would depend on \(x\). "the hourly rate only" is the slope (\(150\)). The \(y\)-intercept here represents the base cost, which can be related to the cost for the first part (before the hourly rate applies fully), and the option "the rate for the first, 20 minutes" (assuming the base cost relates to the initial time) or more accurately, the fixed cost. But from the options, the \(y\)-intercept is the fixed cost, and if we assume the options are about what the \(y\)-intercept represents, and the correct option (from the context) is the one that represents the fixed cost. But looking at the options, the correct one should be the one that represents the cost when \(x = 0\), which is the "the acting - incident after the hourly rate" (probably a typo, and it should be the fixed cost, like the cost before the hourly rate applies, or the base charge). But based on the slope - intercept form, when \(x = 0\), \(y = 50\), so the \(y\)-intercept is the cost when no hours are spent (the base cost). So the correct option is the one that represents the fixed cost, and among the options, the one that is the \(y\)-intercept (the cost at \(x = 0\)).

Answer:

(Assuming the options are as follows, and the correct one is the one representing the \(y\)-intercept (fixed cost). If the options are:

1. the total for the entire visit
2. the hourly rate only
3. the rate for the first, 20 minutes
4. the acting - incident after the hourly rate (fixed cost)

Then the answer is the option representing the \(y\)-intercept, which is the fixed cost, so the option like "the acting - incident after the hourly rate" (assuming it's a typo for the fixed cost) or the correct option based on the slope - intercept analysis. But from the equation \(y = 150x + 50\), the \(y\)-intercept is 50, which is the cost when \(x = 0\) (no hours spent), so it's the base cost. So the correct option is the one that is the \(y\)-intercept, which is the fixed cost, and among the given options, the one that represents the cost at \(x = 0\).)

(Note: There seems to be a typo in the options, but based on the mathematical analysis of the slope - intercept form, the \(y\)-intercept in \(y=150x + 50\) is 50, which is the cost when the number of hours \(x = 0\) (no hours spent at the home), so it's a fixed cost. So the correct option should be the one that represents this fixed cost, which is likely the option "the acting - incident after the hourly rate" (assuming a typo) or the option representing the cost at \(x = 0\).)