Let's write rules for the functions, A(t) and B(t), first and then we can compare their slopes and y-intercepts. Finally, we will determine their domains and ranges and compare those as well.
Rule for A(t)
The box for Pump A claims that it can move 48 gallons of water per minute. Therefore, in t minutes, it will move
48t gallons of water.
Thus, the function A(t), which represents the amount of water Pump A can move after t minutes, can be written as:
A(t)=48t.
Rule for B(t)
We must determine the rule for B(t) by looking at the graph. Let's identify the coordinates of two points on the graph to find the slope first!
Now we can substitute (0,0) and (20,900) into the Slope Formula!
m = y_2 - y_1/x_2 - x_1
m = 900 - 0/20 - 0
m = 900/20
m = 45
The slope of this function is 45. Also, because the graph passes through the origin, we know that the y-intercept is 0. Therefore, we can write the final rule for B(t).
B(t)=45t
Comparison of slopes and y-intercepts
We now have the rules for the two functions, A(t) and B(t).
A(t)&= 48t
B(t)&= 45t
We can see that the slope of A(t) is greater than the slope of B(t). However, both functions have the same y-intercept, y=0.
Domains
A domain is the set of all possible input values for a function. There are no restrictions on the maximum time Pump A can move the water. The minimum amount of water that can be pumped is 0 gallons, the machine hasn't even been turned on. Therefore, the domain is all real values of t which are greater than or equal to 0.
t ≥ 0
From the graph, we can see that the possible values of t for B(t) start from t=0 and end at t=20. However, we do not know what happens after that time, so we can assume that the graph continues on forever. Thus, the domain of B(t) is also all real numbers greater than or equal to 0.
t ≥ 0
Ranges
Since the domain of A(t) is t ≥ 0, its range is the set of all real values greater than A(0) because the slope is positive. However, A(0) is the y-intercept of A(t), which we know is 0. Therefore, the range is:
A(t) ≥ 0.
We can identify the range of B(t) by looking at the outputs of B(t) on the graph. The function starts at B(t)=0, and since we assumed that it will continue on forever, it does not have a maximum value. Its range is:
B(t) ≥ 0.
