Answer :
The temperature of the oven is modeled by the function
$$
f(t)=349.2(0.98)^t,
$$
where:
- $349.2$ is the initial temperature of the oven,
- $0.98$ is the factor by which the temperature is multiplied each minute,
- $t$ is the time in minutes.
We can calculate the temperature at specific times as follows:
1. **At $t=0$ minutes:**
When $t=0$, the temperature is
$$
f(0)=349.2(0.98)^0=349.2.
$$
2. **At $t=10$ minutes:**
When $t=10$, the temperature is
$$
f(10)=349.2(0.98)^{10}\approx285.3218241651313.
$$
3. **At $t=20$ minutes:**
When $t=20$, the temperature is
$$
f(20)=349.2(0.98)^{20}\approx233.1287037368789.
$$
Next, we determine the time required for the oven to cool to $250$ degrees by solving for $t$ in the equation
$$
349.2(0.98)^t=250.
$$
4. **Solving for the time when $f(t)=250$:**
Divide both sides of the equation by $349.2$:
$$
(0.98)^t=\frac{250}{349.2}.
$$
Take the natural logarithm of both sides:
$$
\ln\left((0.98)^t\right)=\ln\left(\frac{250}{349.2}\right).
$$
Use the logarithm power rule:
$$
t\ln(0.98)=\ln\left(\frac{250}{349.2}\right).
$$
Solve for $t$:
$$
t=\frac{\ln\left(\frac{250}{349.2}\right)}{\ln(0.98)}\approx16.54154073763182.
$$
Thus, the computed results are:
- At $t=0$ minutes: Temperature is $349.2$ degrees.
- At $t=10$ minutes: Temperature is approximately $285.3218$ degrees.
- At $t=20$ minutes: Temperature is approximately $233.1287$ degrees.
- The time required for the oven to cool to $250$ degrees is approximately $16.5415$ minutes.
$$
f(t)=349.2(0.98)^t,
$$
where:
- $349.2$ is the initial temperature of the oven,
- $0.98$ is the factor by which the temperature is multiplied each minute,
- $t$ is the time in minutes.
We can calculate the temperature at specific times as follows:
1. **At $t=0$ minutes:**
When $t=0$, the temperature is
$$
f(0)=349.2(0.98)^0=349.2.
$$
2. **At $t=10$ minutes:**
When $t=10$, the temperature is
$$
f(10)=349.2(0.98)^{10}\approx285.3218241651313.
$$
3. **At $t=20$ minutes:**
When $t=20$, the temperature is
$$
f(20)=349.2(0.98)^{20}\approx233.1287037368789.
$$
Next, we determine the time required for the oven to cool to $250$ degrees by solving for $t$ in the equation
$$
349.2(0.98)^t=250.
$$
4. **Solving for the time when $f(t)=250$:**
Divide both sides of the equation by $349.2$:
$$
(0.98)^t=\frac{250}{349.2}.
$$
Take the natural logarithm of both sides:
$$
\ln\left((0.98)^t\right)=\ln\left(\frac{250}{349.2}\right).
$$
Use the logarithm power rule:
$$
t\ln(0.98)=\ln\left(\frac{250}{349.2}\right).
$$
Solve for $t$:
$$
t=\frac{\ln\left(\frac{250}{349.2}\right)}{\ln(0.98)}\approx16.54154073763182.
$$
Thus, the computed results are:
- At $t=0$ minutes: Temperature is $349.2$ degrees.
- At $t=10$ minutes: Temperature is approximately $285.3218$ degrees.
- At $t=20$ minutes: Temperature is approximately $233.1287$ degrees.
- The time required for the oven to cool to $250$ degrees is approximately $16.5415$ minutes.
