Question
A student makes money by watching the neighbors’ dog. The situation is modeled in the graph below.
Point 1: (10, 120)
Point 2: (5, 70)
Which statement correctly describes the relationship between the amount of money the student makes and time in days?
A.
The student charges $11 plus an additional $20 per day.
B.
The student charges $20 plus an additional $11 per day.
C.
The student charges $20 plus an additional $10 per day.
D.
The student charges $10 plus an additional $20 per day.
Point 1: (10, 120)
Point 2: (5, 70)
Which statement correctly describes the relationship between the amount of money the student makes and time in days?
A.
The student charges $11 plus an additional $20 per day.
B.
The student charges $20 plus an additional $11 per day.
C.
The student charges $20 plus an additional $10 per day.
D.
The student charges $10 plus an additional $20 per day.
Answers
GPT-4o mini
To find the correct relationship between the amount of money made and time in days, we can calculate the slope (rate of money earned per day) and the y-intercept (initial charge).
We have the points (5, 70) and (10, 120).
First, we calculate the slope (m) using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using the points (5, 70) as (x1, y1) and (10, 120) as (x2, y2):
\[ m = \frac{120 - 70}{10 - 5} = \frac{50}{5} = 10 \]
This means the student earns $10 for each additional day they watch the dog.
Next, we can find the y-intercept (the initial charge when x = 0, which represents the charge for the first day). We can use one of the points to find the equation of the line. Using point (5, 70):
The formula for the line is:
\[ y = mx + b \]
where \( m = 10 \) and (x, y) is (5, 70):
\[ 70 = 10(5) + b \]
\[ 70 = 50 + b \]
\[ b = 70 - 50 = 20 \]
Thus, the equation that represents the relationship is:
\[ y = 10x + 20 \]
This indicates that the student starts with a charge of $20 and earns an additional $10 for each day.
Now we can match this with the answer choices:
A. The student charges $11 plus an additional $20 per day.
B. The student charges $20 plus an additional $11 per day.
C. The student charges $20 plus an additional $10 per day.
D. The student charges $10 plus an additional $20 per day.
The correct statement is:
**C. The student charges $20 plus an additional $10 per day.**
We have the points (5, 70) and (10, 120).
First, we calculate the slope (m) using the formula:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} \]
Using the points (5, 70) as (x1, y1) and (10, 120) as (x2, y2):
\[ m = \frac{120 - 70}{10 - 5} = \frac{50}{5} = 10 \]
This means the student earns $10 for each additional day they watch the dog.
Next, we can find the y-intercept (the initial charge when x = 0, which represents the charge for the first day). We can use one of the points to find the equation of the line. Using point (5, 70):
The formula for the line is:
\[ y = mx + b \]
where \( m = 10 \) and (x, y) is (5, 70):
\[ 70 = 10(5) + b \]
\[ 70 = 50 + b \]
\[ b = 70 - 50 = 20 \]
Thus, the equation that represents the relationship is:
\[ y = 10x + 20 \]
This indicates that the student starts with a charge of $20 and earns an additional $10 for each day.
Now we can match this with the answer choices:
A. The student charges $11 plus an additional $20 per day.
B. The student charges $20 plus an additional $11 per day.
C. The student charges $20 plus an additional $10 per day.
D. The student charges $10 plus an additional $20 per day.
The correct statement is:
**C. The student charges $20 plus an additional $10 per day.**