Sara works in an ice cream shop. The table shows the number of hours she worked during each of the past four weeks and the amount she earned each week.
Time Worked Amount Earned
(hours) (dollars)
15 127.50
16.5 140.25
12.5 106.25
17 144.50
Let t represent the time in hours and let a represent the amount earned in dollars. Which equation models the relationship between these variables?
To determine the equation that models the relationship between time worked and amount earned, we need to find the equation for a straight line that best fits the given data points.
We can use the slope-intercept form of a linear equation, which is: y = mx + b, where y represents the amount earned (a), x represents the time worked (t), m represents the slope, and b represents the y-intercept.
To calculate the slope (m), we need to find the difference in amounts earned divided by the difference in time worked for any two points. Let's choose the first and fourth data points:
Slope (m) = (Amount Earned difference) / (Time Worked difference)
= (144.50 - 127.50) / (17 - 15)
= 17 / 2
= 8.5
Now, we can substitute one of the data points (e.g., t = 15, a = 127.50) and the calculated slope (m = 8.5) into the slope-intercept form to solve for the y-intercept (b):
127.50 = 8.5 * 15 + b
127.50 = 127.50 + b
b = 0
So the y-intercept (b) is 0.
Now we have the slope (m = 8.5) and y-intercept (b = 0). We can write the equation that models the relationship between time worked (t) and amount earned (a):
a = 8.5t
Thus, the equation that models the relationship between these variables is a = 8.5t.
The equation that models the relationship between time worked and amount earned is:
a = 7.5t + 100
This is a linear equation of the form y = mx + b, where t is the independent variable (time worked), a is the dependent variable (amount earned), m is the slope (rate of pay per hour, which is $7.50), and b is the y-intercept (base pay, which is $100).
The equation shows that for every hour worked, Sara earns $7.50 on top of her base pay of $100. For example, if she works 20 hours, her earnings would be:
a = 7.5(20) + 100
a = 150 + 100
a = 250
Therefore, Sara would earn $250 for working 20 hours.
To find the equation that models the relationship between time worked and amount earned, we need to determine the mathematical relationship between the two variables.
Looking at the given data, we can notice that there is a linear relationship between time worked (t) and amount earned (a).
In a linear equation, the general form is:
a = mt + b
where:
a = amount earned in dollars
t = time worked in hours
m = slope (the rate at which the amount earned changes per hour worked)
b = y-intercept (the amount earned when no time is worked)
To find the values of m and b, we can use any two data points from the table:
Using the first data point (15 hours, $127.50 earned):
127.50 = 15m + b
Using the third data point (12.5 hours, $106.25 earned):
106.25 = 12.5m + b
Now, we have a system of equations. We can solve for m and b by using substitution, elimination, or any other method:
Equation 1: 127.50 = 15m + b
Equation 2: 106.25 = 12.5m + b
By subtracting Equation 2 from Equation 1, we can eliminate the b term:
127.50 - 106.25 = 15m - 12.5m
21.25 = 2.5m
Dividing both sides of the equation by 2.5, we get:
m = 8.5
Now, substitute the value of m into either equation to solve for b:
127.50 = 15(8.5) + b
127.50 = 127.5 + b
By subtracting 127.5 from both sides of the equation:
0 = b
Therefore, the equation that models the relationship between time worked (t) and amount earned (a) is:
a = 8.5t + 0, which simplifies to:
a = 8.5t