College

Catherine placed a turkey in the oven. The temperature was initially [tex]70^{\circ} F[/tex]. After being in the oven for 20 minutes, the temperature was [tex]115^{\circ} F[/tex].

Write a point-slope equation to represent this situation using the formula:

[tex]y = m(x - x_1) + y_1[/tex]

Answer :

To write a point-slope equation for the situation described, let's follow these steps:

1. Identify the Given Information:
- The initial temperature of the turkey when it was placed in the oven: [tex]\( 70^\circ F \)[/tex].
- The temperature after 20 minutes: [tex]\( 115^\circ F \)[/tex].

2. Understand the Question:
- We're asked to write a point-slope equation, which is generally in the form:
[tex]\[
y = m(x - x_1) + y_1
\][/tex]
where [tex]\( m \)[/tex] is the slope, and [tex]\( (x_1, y_1) \)[/tex] is a point on the line.

3. Calculate the Slope ([tex]\( m \)[/tex]):
- The slope [tex]\( m \)[/tex] represents the rate of temperature increase per minute.
- We have two points: [tex]\((0, 70)\)[/tex] and [tex]\((20, 115)\)[/tex].
- 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:
[tex]\[
m = \frac{115 - 70}{20 - 0} = \frac{45}{20} = 2.25
\][/tex]

4. Write the Point-Slope Equation:
- We can use the point [tex]\((20, 115)\)[/tex] in the point-slope form:
[tex]\[
y = m(x - x_1) + y_1
\][/tex]
- Plug in [tex]\( m = 2.25 \)[/tex], [tex]\( x_1 = 20 \)[/tex], and [tex]\( y_1 = 115 \)[/tex]:
[tex]\[
y = 2.25(x - 20) + 115
\][/tex]

Therefore, the point-slope equation representing the situation is:
[tex]\[ y = 2.25(x - 20) + 115 \][/tex]

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