Question

dr. shoh 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?

the rate for the first 20 minutes
the rate for the entire visit
the hourly rate only
the extra amount after the hourly rate

Explanation:

Step1: Recall slope-intercept form

The equation of a line is \( y = mx + b \), where \( m \) is the slope (rate of change) and \( b \) is the y-intercept (value of \( y \) when \( x = 0 \)).

Step2: Analyze the given equation

The given equation is \( y = 150x + 50 \). Here, \( x \) represents the number of hours spent at the home. When \( x = 0 \), it means 0 hours (or 0 minutes) have been spent.

Step3: Interpret the y-intercept

Substitute \( x = 0 \) into the equation: \( y = 150(0) + 50 = 50 \). This means when the time spent (\( x \)) is 0 (i.e., at the start of the visit, before any hours are spent), the cost (\( y \)) is 50. So this should represent the base cost, likely the rate for the initial part of the visit (like the first 20 minutes, as one of the options suggests) or a fixed initial charge. Looking at the options, the y-intercept (50) is a fixed amount when \( x = 0 \) (no hours spent yet), so it's the rate for the first part (like first 20 minutes) or the fixed initial charge. Among the options, "the rate for the first 20 minutes" (or the fixed initial charge which could correspond to the first part of the visit) makes sense. Wait, actually, when \( x = 0 \), it's the cost before any hours, so it's the initial charge, which could be the rate for the first part (like first 20 minutes) since 20 minutes is a fraction of an hour, but when \( x = 0 \), it's the starting cost. So the y-intercept is the value when \( x = 0 \), so it's the cost when 0 hours are spent, which would be the initial charge, like the first 20 minutes (since 20 minutes is not an hour, so \( x = 0 \) would be before any hours, so the initial fixed charge, which is the rate for the first part of the visit). So the correct option is the one about the first 20 minutes or the initial charge. Wait, the options are: "the rate for the first 20 minutes", "the rate for the entire visit", "the hourly rate only", "the extra amount after the hourly rate". The y-intercept is \( b = 50 \), which is a fixed cost when \( x = 0 \) (0 hours). So when \( x = 0 \), it's the cost before any hours, so it's the initial charge, which is the rate for the first part (like first 20 minutes) because after that, the hourly rate (slope \( 150 \)) applies. So the y-intercept is the cost when \( x = 0 \), so it's the rate for the first 20 minutes (or the initial fixed charge). So the correct option is "the rate for the first 20 minutes".

Answer:

The option "the rate for the first 20 minutes" (assuming the options are labeled, but from the image, the first option on the right is "the rate for the first 20 minutes", so the answer is that option. If we list the options as per the image (from right to left? Wait, the image has four options: first (rightmost) is "the rate for the first 20 minutes", then "the rate for the entire visit", then "the hourly rate only", then "the extra amount after the hourly rate". So the correct option is the first one (rightmost) which is "the rate for the first 20 minutes".