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Answer :
Let's start with the information given in the problem. We know the following:
- At 2 minutes, the oven's temperature is 180°F.
- At 4 minutes, the oven's temperature is 290°F.
We can use these two points to find the equation of the line that represents the temperature of the oven as a function of time. The points are (2, 180) and (4, 290).
To find this equation, we need to determine the slope (m) and the y-intercept (b) of the line.
### Step 1: Calculate the Slope (m)
The formula for the slope between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substitute the given points (2, 180) and (4, 290):
[tex]\[ m = \frac{290 - 180}{4 - 2} \][/tex]
[tex]\[ m = \frac{110}{2} \][/tex]
[tex]\[ m = 55 \][/tex]
### Step 2: Find the y-intercept (b)
The equation of the line in slope-intercept form is:
[tex]\[ y = mx + b \][/tex]
We already know [tex]\(m = 55\)[/tex]. To find the y-intercept [tex]\(b\)[/tex], we can use either of the two points given. Let's use the point (2, 180):
[tex]\[ 180 = 55(2) + b \][/tex]
[tex]\[ 180 = 110 + b \][/tex]
[tex]\[ b = 180 - 110 \][/tex]
[tex]\[ b = 70 \][/tex]
### Step 3: Write the Equation
Now that we have the slope and the y-intercept, we can write the equation of the line:
[tex]\[ y = 55x + 70 \][/tex]
So, the equation to represent [tex]\(y\)[/tex], the temperature of the oven, after [tex]\(x\)[/tex] minutes is:
[tex]\[ y = 55x + 70 \][/tex]
- At 2 minutes, the oven's temperature is 180°F.
- At 4 minutes, the oven's temperature is 290°F.
We can use these two points to find the equation of the line that represents the temperature of the oven as a function of time. The points are (2, 180) and (4, 290).
To find this equation, we need to determine the slope (m) and the y-intercept (b) of the line.
### Step 1: Calculate the Slope (m)
The formula for the slope between two points [tex]\((x_1, y_1)\)[/tex] and [tex]\((x_2, y_2)\)[/tex] is:
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substitute the given points (2, 180) and (4, 290):
[tex]\[ m = \frac{290 - 180}{4 - 2} \][/tex]
[tex]\[ m = \frac{110}{2} \][/tex]
[tex]\[ m = 55 \][/tex]
### Step 2: Find the y-intercept (b)
The equation of the line in slope-intercept form is:
[tex]\[ y = mx + b \][/tex]
We already know [tex]\(m = 55\)[/tex]. To find the y-intercept [tex]\(b\)[/tex], we can use either of the two points given. Let's use the point (2, 180):
[tex]\[ 180 = 55(2) + b \][/tex]
[tex]\[ 180 = 110 + b \][/tex]
[tex]\[ b = 180 - 110 \][/tex]
[tex]\[ b = 70 \][/tex]
### Step 3: Write the Equation
Now that we have the slope and the y-intercept, we can write the equation of the line:
[tex]\[ y = 55x + 70 \][/tex]
So, the equation to represent [tex]\(y\)[/tex], the temperature of the oven, after [tex]\(x\)[/tex] minutes is:
[tex]\[ y = 55x + 70 \][/tex]
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