\( T \) represents the temperature in \( t \) minutes. \( T=42-0.7 t \)

Certainly! You've provided a linear equation that describes how temperature \( T \) changes over time \( t \) in minutes: \[ T = 42 - 0.7t \] Let's break down and analyze this equation to understand its implications. ## Components of the Equation 1. **Variables:** - \( T \): Temperature at time \( t \) (units likely in degrees, such as Celsius or Fahrenheit). - \( t \): Time in minutes. 2. **Constants:** - **Initial Temperature (Intercept):** 42 - When \( t = 0 \), \( T = 42 \). This is the starting temperature. - **Rate of Change (Slope):** -0.7 - The temperature decreases by 0.7 degrees every minute. ## Interpretation - **Linear Relationship:** The equation is linear, indicating a constant rate of temperature change over time. - **Cooling Process:** Given the negative slope, this model likely represents a cooling process where the temperature decreases consistently as time progresses. ## Key Features and Calculations 1. **Initial Temperature (\( t = 0 \)):** \[ T(0) = 42 - 0.7 \times 0 = 42 \text{ degrees} \] - The system starts at 42 degrees. 2. **Rate of Cooling:** - Every minute, the temperature drops by 0.7 degrees. 3. **Time to Reach a Specific Temperature:** - **Example:** When will the temperature reach 0 degrees? \[ 0 = 42 - 0.7t \\ 0.7t = 42 \\ t = \frac{42}{0.7} = 60 \text{ minutes} \] - It will take 60 minutes for the temperature to decrease from 42 degrees to 0 degrees. 4. **Graphical Representation:** - **Y-Intercept:** (0, 42) – Starting point on the temperature axis. - **Slope:** -0.7 – Indicates the steepness and direction (downward slope). - **X-Intercept:** (60, 0) – Point where temperature reaches 0 degrees.  *(Replace the link with an actual graph if needed.)* ## Possible Applications - **Cooling of Liquids:** Such as cooling coffee or other beverages over time. - **Environmental Studies:** Modeling how temperature decreases in a shaded area or during evening hours. - **Engineering:** Assessing how a device or system cools down after shutdown. ## Additional Considerations - **Real-World Factors:** While the model assumes a constant rate of temperature change, real-world scenarios might experience variable cooling rates due to factors like ambient temperature, insulation, or material properties. - **Time Frame:** Ensure that the model's application doesn't extend beyond its valid range. For instance, predicting negative temperatures beyond physical plausibility. ## Summary The equation \( T = 42 - 0.7t \) provides a straightforward linear model to describe how temperature decreases over time at a constant rate of 0.7 degrees per minute, starting from an initial temperature of 42 degrees. This can be useful for making predictions about when the temperature will reach certain values or understanding the rate at which cooling occurs in the modeled scenario. If you have a specific question or need further analysis related to this equation, feel free to ask!