Question

yuson must complete a certain number of hours of community service. she does 2 hours of service each day. after 8 days, yuson has completed half of her required community service. for the graph of the equation that represents this situation, what does the y - intercept represent?
a hours left to complete
b total hours of service
c hours completed each day
d days it takes to complete 30 hours

Explanation:

Step1: Analyze the given information

Yuson does 2 hours of service each day. After 8 days, she has completed half of her required community service. First, find the number of hours completed in 8 days: \( 2 \times 8 = 16 \) hours. Since this is half of the total required, the total hours of service (let's call it \( T \)) is \( 16 \times 2 = 32 \)? Wait, no, wait. Wait, the problem says "after 8 days, Yuson has completed half of her required community service". Wait, no, maybe I misread. Wait, let's re-express. Let's define the equation. Let \( x \) be the number of days, \( y \) be the number of hours completed. The rate is 2 hours per day, so the equation is \( y = 2x + b \)? Wait, no, the y - intercept is when \( x = 0 \), which is the initial number of hours, but wait, maybe the total hours. Wait, after 8 days, she has completed half. So in 8 days, she does \( 2\times8 = 16 \) hours, which is half of the total. So total hours \( T = 32 \)? Wait, no, maybe the equation is about the hours left. Wait, the question is about the y - intercept of the graph representing the situation. Let's think about the variables. Let's suppose the equation is for the hours left to complete. Let \( y \) be hours left, \( x \) be days. The rate of completing is 2 hours per day, so the hours left decrease by 2 each day. After 8 days, she has completed half, so hours left after 8 days is equal to hours completed (since half done, half left). Hours completed in 8 days: \( 2\times8 = 16 \), so hours left after 8 days is 16. So the equation for hours left: \( y = - 2x + 32 \) (since total hours is 32, as 16 done and 16 left after 8 days, so total is 32). Wait, but the y - intercept is when \( x = 0 \), which is 32, the total hours? No, wait, if the equation is for hours left, when \( x = 0 \) (no days passed), hours left is total hours. But the options: B is total hours of service. Wait, let's check the options again.

Wait, the problem says "Yuson must complete a certain number of hours of community service. She does 2 hours of service each day. After 8 days, Yuson has completed half of her required community service." So in 8 days, she does \( 2\times8 = 16 \) hours, which is half of the total. So total hours \( T = 16\times2 = 32 \). Now, if we model the total hours completed as \( y = 2x \), but that would have y - intercept 0, which is not an option. If we model the hours left, \( y = T - 2x = 32 - 2x \). The y - intercept is when \( x = 0 \), so \( y = 32 \), which is the total hours of service (option B). Wait, but let's check the options:

A: hours left to complete (y - intercept would be total hours, which is the initial hours left, but if the equation is hours left, then yes, but let's see. Wait, maybe the equation is for total hours completed. No, total hours completed at x = 0 is 0. If the equation is for hours left, then y - intercept is total hours. So the y - intercept represents the total hours of service (option B).

Wait, but let's re - evaluate. Let's consider the equation of the line. Let's define the variables. Let’s say the graph is of the total hours of service? No, total hours is constant. Wait, maybe the equation is \( y = 2x + b \), but that doesn't make sense. Wait, the key is: when x = 0 (no days worked), what does y represent? If we are talking about the total hours required, then when x = 0, the total hours is the y - intercept. Since after 8 days, she has done 16 hours (half), total is 32. So the y - intercept (when x = 0) would be the total hours of service, which is option B.

Step2: Evaluate the options

- Option A: hours left to complete. If the equation is for hours left, the y - intercept is total hours, not hours left at x = 0 (wait, at x = 0, hours left is total hours, but if the equation is \( y=-2x + 32 \), then y - intercept is 32, which is total hours, not hours left in the sense of "left" as in remaining, but total. Wait, maybe I confused. Let's think again. The total hours of service is the total she needs to do. When she starts (x = 0), the total hours is the y - intercept. So option B is correct.
- Option C: hours completed each day is 2, which is the slope, not the y - intercept.
- Option D: days to complete 30 hours is not related to the y - intercept.
- Option A: hours left to complete at x = 0 would be total hours, but the option says "hours left to complete", but the y - intercept is the total hours, which is the initial hours left (before any days), but the total hours is the total she needs to do, so option B is total hours of service.

Answer:

B. total hours of service