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The function [tex]f(t)=349.2(0.98)^t[/tex] models the relationship between [tex]t[/tex], the time an oven spends cooling, and the temperature of the oven. Describe how the temperature changes over time.

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.

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