Question

a pan of brownies is taken out of the oven. the temperature of the brownies, in degrees celsius, is given by the function $t(x) = 155e^{-0.0943x} + 20$, where $x$ is the time in minutes since the brownies were removed from the oven and $t$ is the temperature.
a. what is the temperature of the brownies when they are first removed from the oven? round to the nearest degree.
b. what is the temperature of the brownies 10 minutes after they are removed from the oven? round to the nearest degree.
c. what temperature will the brownies eventually cool to? round to the nearest degree.

Explanation:

Part (a)

Step1: Identify \( x \) for first removal

When first removed, \( x = 0 \) (time since removal is 0 minutes).

Step2: Substitute \( x = 0 \) into \( T(x) \)

\( T(0)=155e^{-0.0943(0)} + 20 \)
Since \( e^{0}=1 \), this becomes \( 155(1)+20 = 155 + 20 = 175 \).

Step1: Identify \( x \) for 10 minutes

For 10 minutes after removal, \( x = 10 \).

Step2: Substitute \( x = 10 \) into \( T(x) \)

\( T(10)=155e^{-0.0943(10)} + 20 \)
First calculate \( -0.0943(10)= - 0.943 \), then \( e^{-0.943}\approx0.3907 \).
Then \( 155\times0.3907\approx60.56 \), and \( 60.56 + 20 = 80.56\approx81 \) (rounded to nearest degree).

Step1: Analyze limit as \( x\to\infty \)

As \( x \) approaches infinity, \( e^{-0.0943x}=\frac{1}{e^{0.0943x}}\to0 \) (since exponential function with positive exponent grows to infinity, so its reciprocal goes to 0).

Step2: Find limit of \( T(x) \) as \( x\to\infty \)

\( \lim_{x\to\infty}T(x)=\lim_{x\to\infty}(155e^{-0.0943x}+20)=155(0)+20 = 20 \).

Answer:

175

Part (b)