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As a certain engine's rotation speed increases, its temperature increases at a constant rate.
The table compares the engine's rotation speed (in cycles per second) and its temperature (in degrees Celsius).
Rotation speed (cycles per second)	Temperature (degrees Celsius)
\[9\]	\[22.2\]
\[12\]	\[24.6\]
\[15\]	\[27.0\]
What is the engine's temperature at rest?

degrees Celsius

To find the engine's temperature at rest (i.e., when the rotation speed is 0 cycles per second), we can assume that the relationship between the rotation speed and temperature is linear, based on the given information.

We are given three data points:
- At 9 cycles/second, the temperature is 22.2°C.
- At 12 cycles/second, the temperature is 24.6°C.
- At 15 cycles/second, the temperature is 27.0°C.

### Step 1: Determine the rate of temperature change
Since the rate of temperature increase is constant, we calculate the rate (or slope) of this linear relationship:

\[
\text{slope} = \frac{\Delta T}{\Delta R} = \frac{27.0 - 22.2}{15 - 9} = \frac{4.8}{6} = 0.8 \, \text{degrees per cycle/second}
\]

### Step 2: Use the slope to find the y-intercept
We can now use the point-slope form of the equation for a line:

\[
T = mR + b
\]

Where:
- \(T\) is the temperature,
- \(R\) is the rotation speed,
- \(m\) is the slope (0.8), and
- \(b\) is the y-intercept (the temperature at rest, when \(R = 0\)).

Let's use one of the points, say (9, 22.2), to find \(b\):

\[
22.2 = 0.8(9) + b
\]

\[
22.2 = 7.2 + b
\]

\[
b = 22.2 - 7.2 = 15.0
\]

### Step 3: Conclusion
The y-intercept is 15.0°C. Therefore, the engine's temperature at rest (0 cycles per second) is **15.0 degrees Celsius**.

Would you like more details or have any questions? Here are some related questions:

1. How do we determine if a relationship is linear?
2. What happens if the temperature increases at a non-constant rate?
3. How can we model other real-world phenomena with linear equations?
4. What are the potential factors affecting the temperature of an engine beyond speed?
5. How does the slope of a line relate to real-world rates of change?

**Tip:** Always check the units of measurement when solving real-world problems to avoid confusion!

As a certain engine's rotation speed increases, its temperature increases at a constant rate. The table compares the engine's rotation speed (in cycles per second) and its temperature (in degrees Celsius). What is the engine's temperature at rest?

Discover how to calculate the engine's temperature at rest by analyzing its linear relationship between rotation speed and temperature. Learn the key steps, including calculating the slope and determining the y-intercept, based on a real-world scenario. Suitable for students in Grades 9-11.

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