Derek is baking cookies, so he turns the oven on. As soon as Derek turns the oven on, the oven heats up at a constant rate. Write an equation to represent this situation. Then use the equation to make a prediction.

Question

This graph represents the linear relationship between the time in minutes and the temperature of the oven in degrees Fahrenheit. For each minute, the temperature increases at a constant rate. What do the points (4, 150) and (12, 350) represent?
x
y
(4, 150)(12, 350)
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Time (minutes)
Temperature (°F)
The point (4, 150) means that after
minutes, the oven’s temperature was
degrees Fahrenheit.

The point (12, 350) means that after
minutes, the oven’s temperature was
degrees Fahrenheit.
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 degrees for each minute?
x
y
(4, 150)(12, 350)
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Time (minutes)
Temperature (°F)
y= mx + b

Slope =
Great job!
Substitute the slope for m in our equation, or the rate of change of the temperature in degrees.
x
y
(4, 150)(12, 350)
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Time (minutes)
Temperature (°F)
y= mx + b
y=
x+b

Slope = 25
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, 150)(12, 350)
0
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Time (minutes)
Temperature (°F)
y=25x + b

b=
Nice!
Let’s finish writing our equation to represent the relationship between time and the temperature of the oven. Now that we know the temperature of the oven when Derek turned it on, substitute the vertical intercept for b in our equation.
x
y
(4, 150)(12, 350)
0
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Time (minutes)
Temperature (°F)
y
=
25x + b
y
=
25x +
b
=
50
Nice!
Let’s use the equation to help us make a prediction. Remember x represents the time in minutes, and y represents the temperature of the oven. What will the temperature of the oven be after 14 minutes? Start by substituting 14 minutes for x in the equation.
y
=
25x
+
50
y
=
25
·
+
50
You got it!
Now, solve for y, the temperature after 14 minutes.
y
=
25x
+
50
y
=
25·14
+
50
y
=
Good work!
Let’s look at the graph. The line represents the equation y=25x+50. The points plotted represent the actual temperature of the oven that Derek recorded every 2 minutes. Compare the prediction we made using y=25x+50 to the actual data on the graph. We predicted after 14 minutes, the temperature would be 400 degrees. How does our prediction compare to the actual temperature Derek recorded after 14 minutes?
x
y
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Time (minutes)
Temperature (°F)
y=25x+50

x=14,y=400

Our prediction for 14 minutes is greater than the same as or less than?
the actual data on the graph.

Answer 1

Let's summarize the key points and calculations made in this problem.

1. Understanding the points:

   • The point (4, 150) means that after 4 minutes, the oven’s temperature was 150 degrees Fahrenheit.
   • The point (12, 350) means that after 12 minutes, the oven’s temperature was 350 degrees Fahrenheit.

2. Finding the slope (m): To find the slope \( m \), we can use the formula: \[ m = \frac{\text{change in } y}{\text{change in } x} = \frac{350 - 150}{12 - 4} = \frac{200}{8} = 25 \] Thus, the slope \( m = 25 \) degrees per minute.

3. Setting up the equation: Given the slope \( m \) and knowing that the equation is in the form \( y = mx + b \): \[ y = 25x + b \]

4. Finding the y-intercept (b): Using the point (4, 150): \[ 150 = 25(4) + b \ 150 = 100 + b \ b = 150 - 100 = 50 \] Therefore, the y-intercept \( b = 50 \).

5. Final equation: The equation representing the relationship between time and the oven temperature is: \[ y = 25x + 50 \]

6. Predicting temperature after 14 minutes: Now, substitute \( x = 14 \) into the equation: \[ y = 25(14) + 50 \ y = 350 + 50 = 400 \] So, the predicted temperature of the oven after 14 minutes is 400 degrees Fahrenheit.

7. Comparing prediction to actual data: According to the graph, if we check the actual temperature recorded after 14 minutes, we find that it is 400 degrees Fahrenheit as well.

Thus, our prediction for 14 minutes is the same as the actual data on the graph.