Question: The temperature of a cup of coffee can be modeled using the equation $T = -5t + 200$, where $T$ represents the temperature of the coffee in degrees Fahrenheit, and $t$ is the time in minutes since the coffee was poured.
• The ____ intercept would be the initial temperature of the coffee.
• The ____ intercept represents the number of minutes until the coffee cools to room temperature.
• The slope indicates that the coffee's temperature is ____ at 5°F/min.
1. **State the problem:** We have the temperature model $$T = -5t + 200$$ where $T$ is temperature in °F and $t$ is time in minutes.
2. **Identify intercepts:**
- The $t$-intercept is where $T=0$.
- The $T$-intercept is where $t=0$.
3. **Find the $T$-intercept (initial temperature):**
Substitute $t=0$:
$$T = -5(0) + 200 = 200$$
So, the $T$-intercept is $200$°F, which is the initial temperature of the coffee.
4. **Find the $t$-intercept (time when coffee cools to room temperature):**
Set $T=0$ and solve for $t$:
$$0 = -5t + 200$$
$$5t = 200$$
$$t = \frac{200}{5}$$
$$t = 40$$
So, the coffee cools to room temperature after $40$ minutes.
5. **Interpret the slope:**
The slope is $-5$, meaning the temperature decreases by $5$°F every minute.
**Final answers:**
- The $T$-intercept is the initial temperature of the coffee: $200$°F.
- The $t$-intercept represents the number of minutes until the coffee cools to room temperature: $40$ minutes.
- The slope indicates the coffee's temperature decreases at $5$°F per minute.