Answer :
We know that the oven's temperature decreases by a constant amount every minute. This is a key characteristic of a linear function, which involves a constant rate of change (or constant difference). Here’s a step-by-step breakdown:
1. Establish the situation:
The cake was baked in an oven at an initial temperature of [tex]$350^\circ F$[/tex]. After the oven is turned off, the temperature decreases by [tex]$10^\circ F$[/tex] every minute.
2. Identify the type of function:
Because the temperature drops by the same amount ([tex]$10^\circ F$[/tex]) each minute, the function is linear. This is because a linear function has a constant difference between its values (also known as a constant rate of change).
3. Determine the common relationship:
The constant change (decrease) in temperature is a constant difference. Hence, the relationship has a common difference.
4. Construct the function equation:
If we let [tex]$x$[/tex] be the number of minutes after the oven is turned off, then the temperature [tex]$f(x)$[/tex] at time [tex]$x$[/tex] can be modeled by the linear equation:
[tex]$$f(x) = -10x + 350$$[/tex]
Here, [tex]$-10$[/tex] represents the rate at which the temperature decreases per minute (the slope), and [tex]$350$[/tex] is the starting temperature (the y-intercept).
5. Final Answers:
- The scenario can be modeled by a linear function.
- The relationship has a common difference.
- The function that best models the scenario is: [tex]$$f(x) = -10x + 350.$$[/tex]
This completes the step-by-step reasoning for the problem.
1. Establish the situation:
The cake was baked in an oven at an initial temperature of [tex]$350^\circ F$[/tex]. After the oven is turned off, the temperature decreases by [tex]$10^\circ F$[/tex] every minute.
2. Identify the type of function:
Because the temperature drops by the same amount ([tex]$10^\circ F$[/tex]) each minute, the function is linear. This is because a linear function has a constant difference between its values (also known as a constant rate of change).
3. Determine the common relationship:
The constant change (decrease) in temperature is a constant difference. Hence, the relationship has a common difference.
4. Construct the function equation:
If we let [tex]$x$[/tex] be the number of minutes after the oven is turned off, then the temperature [tex]$f(x)$[/tex] at time [tex]$x$[/tex] can be modeled by the linear equation:
[tex]$$f(x) = -10x + 350$$[/tex]
Here, [tex]$-10$[/tex] represents the rate at which the temperature decreases per minute (the slope), and [tex]$350$[/tex] is the starting temperature (the y-intercept).
5. Final Answers:
- The scenario can be modeled by a linear function.
- The relationship has a common difference.
- The function that best models the scenario is: [tex]$$f(x) = -10x + 350.$$[/tex]
This completes the step-by-step reasoning for the problem.
