Leo fills his boat with gas. The boat has some gas in it, and Leo fills it at a constant rate each minute. Write an equation that represents this situation. Then use the equation to make a prediction.

Question

This graph represents the linear relationship between the time in minutes and the gallons of gas in the boat. For each minute, the gallons of gas increases at a constant rate. What do the points (4, 50) and (10, 80) represent?
x
y
(4, 50)(10, 80)
0
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0
10
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30
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Time (minutes)
Gallons of gas
The point (4, 50) means that after
minutes, there are
gallons of gas in the boat.

The point (10, 80) means that after
minutes, there are
gallons of gas in the boat.
Great job!
Let’s write an equation to represent this line in the form y=mx+b. Start by finding m, or the slope. What is the slope of this line? In other words, what is the change in gallons of gas for each minute?
x
y
(4, 50)(10, 80)
0
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0
10
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Time (minutes)
Gallons of gas
y= mx + b

Slope =
Excellent!
Substitute the slope for m in our equation, or the rate of change of the gallons of gas.
x
y
(4, 50)(10, 80)
0
2
4
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18
20
0
10
20
30
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50
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Time (minutes)
Gallons of gas
y= mx + b
y=
x+b

Slope = 5
Great job!
Now, solve for b, or the vertical intercept. You can can substitute the x and y-values for a point on the line into the equation to help you. Solve on paper or use the calculator. Then, enter your answer on Zearn.
x
y
(4, 50)(10, 80)
0
2
4
6
8
10
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18
20
0
10
20
30
40
50
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130
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Time (minutes)
Gallons of gas
y=5x + b

b=
Excellent!
Let’s finish writing our equation to represent the relationship between time and the gallons of gas in the boat. Now that we know the amount of gas in the boat when Leo started filling it, b, substitute the vertical intercept for b in our equation.
x
y
(4, 50)(10, 80)
0
2
4
6
8
10
12
14
16
18
20
0
10
20
30
40
50
60
70
80
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100
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130
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Time (minutes)
Gallons of gas
y
=
5x + b
y
=
5x +
b
=
30
Good work!
Let’s use the equation to help us make a prediction. Remember x represents the time in minutes, and y represents the gallons of gas in the boat. How many gallons of gas will be in the boat after 20 minutes? Start by substituting 20 minutes for x in the equation.
y
=
5x
+
30
y
=
5
·
+
30
Good work!
Now, solve for y, the gallons of gas in the boat after 20 minutes.
y
=
5x
+
30
y
=
5·20
+
30
y
=
Excellent!
Let’s look at the graph. The points plotted represent the actual number of gallons that Leo recorded every 2 minutes. Compare the prediction we made using y=5x+30 to the actual data on the graph. We predicted that after 20 minutes, there would be 130 gallons of gas in the boat. How does our prediction compare to the actual number of gallons Leo recorded after 20 minutes?
x
y
0
2
4
6
8
10
12
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18
20
0
10
20
30
40
50
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130
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Time (minutes)
Gallons of gas
y=5x+30

x=20,y=130

Our prediction for 20 minutes is ?
the actual data on the graph.
A. Less than
B.greater than
C. the same as

Answer 1

Let's break down the problem step by step and confirm everything mentioned:

1. Understanding the Points:

   • The point (4, 50) means that after 4 minutes, there are 50 gallons of gas in the boat.
   • The point (10, 80) means that after 10 minutes, there are 80 gallons of gas in the boat.

2. Finding the Slope (m): To find the slope (m), we can use the slope formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Using the points (4, 50) and (10, 80): \[ m = \frac{80 - 50}{10 - 4} = \frac{30}{6} = 5 \]

3. Writing the Equation: We can now use the slope to form the equation: \[ y = mx + b \] Substituting the slope (m = 5): \[ y = 5x + b \]

4. Finding the y-intercept (b): We will use a point to find b. Using the point (4, 50): \[ 50 = 5(4) + b \] \[ 50 = 20 + b \] \[ b = 50 - 20 = 30 \]

5. Final Equation: Thus, the equation representing the relationship is: \[ y = 5x + 30 \]

6. Making a Prediction: To predict how many gallons of gas will be in the boat after 20 minutes (x = 20): \[ y = 5(20) + 30 \] \[ y = 100 + 30 = 130 \]

7. Comparison with Graph Data: According to our prediction, after 20 minutes, there will be 130 gallons of gas in the boat. To compare this with the actual data on the graph:

   • If the actual data from the graph at x = 20 matches our prediction (130 gallons), the answer would be C. the same as.
   • If it is less than our prediction, the answer would be A. Less than.
   • If it is greater than our prediction, the answer would be B. greater than.

Conclusion: If our predicted number of gallons is indeed confirmed by the graph data to be 130 gallons at 20 minutes, then the answer is C: the same as. If it differs, you would select the appropriate option based on the graph.