Question
Gilberto is baking brownies, so he turns the oven on. As soon as Gilberto 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.
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 (2, 100) and (6, 220) represent?
x
y
(2, 100)(6, 220)
0
2
4
6
8
10
12
14
16
0
20
40
60
80
100
120
140
160
180
200
220
240
260
280
300
320
340
360
Time (minutes)
Temperature (°F)
The point (2, 100) means that after
minutes, the oven’s temperature was
degrees Fahrenheit.
The point (6, 220) means that after
minutes, the oven’s temperature was
degrees Fahrenheit.
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 (2, 100) and (6, 220) represent?
x
y
(2, 100)(6, 220)
0
2
4
6
8
10
12
14
16
0
20
40
60
80
100
120
140
160
180
200
220
240
260
280
300
320
340
360
Time (minutes)
Temperature (°F)
The point (2, 100) means that after
minutes, the oven’s temperature was
degrees Fahrenheit.
The point (6, 220) means that after
minutes, the oven’s temperature was
degrees Fahrenheit.
Answers
GPT-4o mini
To represent the relationship between the time (in minutes) and the temperature (in degrees Fahrenheit) in an equation, we can use the linear equation of the form:
\[ y = mx + b \]
where:
- \( y \) is the temperature,
- \( x \) is the time in minutes,
- \( m \) is the rate of change (temperature increase per minute),
- \( b \) is the initial temperature of the oven (when \( x = 0 \)).
From the points given, we can determine the rate of change (\( m \)). We see that:
- At \( (2, 100) \): After 2 minutes, the temperature is 100°F.
- At \( (6, 220) \): After 6 minutes, the temperature is 220°F.
To find the rate of change, we calculate the slope \( m \):
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{220 - 100}{6 - 2} = \frac{120}{4} = 30
\]
Thus, the oven heats up at a rate of 30°F per minute.
Since we do not have the initial temperature (\( b \)), we can use one of the points to find it. Let's use the point \( (2, 100) \):
\[
100 = 30(2) + b
\]
\[
100 = 60 + b
\]
\[
b = 100 - 60 = 40
\]
So, the equation representing the relationship is:
\[
y = 30x + 40
\]
Now, regarding the points:
- The point \( (2, 100) \) means that after **2 minutes**, the oven’s temperature was **100 degrees Fahrenheit**.
- The point \( (6, 220) \) means that after **6 minutes**, the oven’s temperature was **220 degrees Fahrenheit**.
Using the equation, we can make a prediction. For example, to predict the temperature after 10 minutes:
\[
y = 30(10) + 40 = 300 + 40 = 340
\]
Thus, after 10 minutes, the predicted temperature of the oven would be **340 degrees Fahrenheit**.
\[ y = mx + b \]
where:
- \( y \) is the temperature,
- \( x \) is the time in minutes,
- \( m \) is the rate of change (temperature increase per minute),
- \( b \) is the initial temperature of the oven (when \( x = 0 \)).
From the points given, we can determine the rate of change (\( m \)). We see that:
- At \( (2, 100) \): After 2 minutes, the temperature is 100°F.
- At \( (6, 220) \): After 6 minutes, the temperature is 220°F.
To find the rate of change, we calculate the slope \( m \):
\[
m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{220 - 100}{6 - 2} = \frac{120}{4} = 30
\]
Thus, the oven heats up at a rate of 30°F per minute.
Since we do not have the initial temperature (\( b \)), we can use one of the points to find it. Let's use the point \( (2, 100) \):
\[
100 = 30(2) + b
\]
\[
100 = 60 + b
\]
\[
b = 100 - 60 = 40
\]
So, the equation representing the relationship is:
\[
y = 30x + 40
\]
Now, regarding the points:
- The point \( (2, 100) \) means that after **2 minutes**, the oven’s temperature was **100 degrees Fahrenheit**.
- The point \( (6, 220) \) means that after **6 minutes**, the oven’s temperature was **220 degrees Fahrenheit**.
Using the equation, we can make a prediction. For example, to predict the temperature after 10 minutes:
\[
y = 30(10) + 40 = 300 + 40 = 340
\]
Thus, after 10 minutes, the predicted temperature of the oven would be **340 degrees Fahrenheit**.