Chapter 11.3, Problem 9E

(a)

To determine

To find: The temperature of the liquid after 15 minutes.

Answer to Problem 9E

The temperate of the liquid is $78.7°\text{F}$ .

Explanation of Solution

Given:

The given model is $y=a{e}^{-kt}+c$

The initial temperature of the liquid is $160°\text{F}$ .

The final temperature is $76°\text{F}$ .

The value of $k=0.23$ .

Calculation:

Consider the given model is,

  $y=a{e}^{-kt}+c$

Then,

  $\begin{array}{c}y=\left(160°-76°\right)e^{-\left(0.23\right)\left(15\right)}+76\\ =2.666+76\\ =78.7°\text{F}\end{array}$

Thus, the temperate of the liquid is $78.7°\text{F}$ .

(b)

To determine

To find: Whether the coffee will be too hot or too cold.

Answer to Problem 9E

After 5 minutes the temperature is $90°\text{F}$ that is too cold.

Explanation of Solution

Given:

The desired temperature is $135°\text{F}$ .

The initial temperature is $170°\text{F}$ and the final temperate is $72°\text{F}$ .

The wait time is of 4 minute and the value of k is 0.34.

Calculation:

Consider the given model is,

  $y=a{e}^{-kt}+c$

Then,

  $\begin{array}{c}y=\left(170°-72°\right)e^{-\left(0.34\right)\left(5\right)}+72\\ =17.90+76\\ =90°\text{F}\end{array}$

Thus, after 5 minutes the temperature is $90°\text{F}$ that is too cold.