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t (minutes)
0
4
9
15
20

W(t) (degrees Fahrenheit)
55.0
57.1
61.8
67.9
71.0

The temperature of water in a tub at time
t is modeled by a strictly increasing, twice-differentiable function
W, where
W(t) is measured in degrees Fahrenheit and
t is measured in minutes. At time
t=0, the temperature of the water is
55^(@)F. The water is heated for 30 minutes, beginning at time
t=0. Values of
W(t) at selected times
t for the first 20 minutes are given in the table above.
(a) Use the data in the table to estimate
W^(')(12). Show the computations that lead to your answer. Using correct units, interpret the meaning of your answer in the context of this problem.

\begin{tabular}{|c|c|c|c|c|c|} $\newline$ \hline $t$ (minutes) & $0$ & $4$ & $9$ & $15$ & $20$ \\ $\newline$ \hline $W(t)$ (degrees Fahrenheit) & $55$ . $0$ & $57$ . $1$ & $61$ . $8$ & $67$ . $9$ & $71$ . $0$ \\ $\newline$ \hline $\newline$ \end{tabular} $\newline$ The temperature of water in a tub at time $t$ is modeled by a strictly increasing, twice-differentiable function $W$ , where $W(t)$ is measured in degrees Fahrenheit and $t$ is measured in minutes. At time $t=0$ , the temperature of the water is $55^{\circ} \mathrm{F}$ . The water is heated for $30$ minutes, beginning at time $t=0$ . Values of $W(t)$ at selected times $t$ for the first $20$ minutes are given in the table above. $\newline$ (a) Use the data in the table to estimate $W(t)$ $1$ . Show the computations that lead to your answer. Using correct units, interpret the meaning of your answer in the context of this problem.

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Evaluate two-variable equations: word problems

Full solution

Q. \begin{tabular}{|c|c|c|c|c|c|}

\newline

\hline

t

(minutes) &

0

&

4

&

9

&

15

&

20

\\

\newline

\hline

W(t)

(degrees Fahrenheit) &

55

.

0

&

57

.

1

&

61

.

8

&

67

.

9

&

71

.

0

\\

\newline

\hline

\newline

\end{tabular}

\newline

The temperature of water in a tub at time

t

is modeled by a strictly increasing, twice-differentiable function

W

, where

W(t)

is measured in degrees Fahrenheit and

t

is measured in minutes. At time

t=0

, the temperature of the water is

55^{\circ} \mathrm{F}

. The water is heated for

30

minutes, beginning at time

t=0

. Values of

W(t)

at selected times

t

for the first

20

minutes are given in the table above.

\newline

(a) Use the data in the table to estimate

W(t)

1

. Show the computations that lead to your answer. Using correct units, interpret the meaning of your answer in the context of this problem.

Estimate $W'(12)$ : To estimate $W'(12)$ , we'll use the average rate of change between two points close to $t=12$ . We can use the interval from $t=9$ to $t=15$ .
Calculate average rate of change: Calculate the average rate of change using the formula: $(W(t_2) - W(t_1)) / (t_2 - t_1)$ . Here, $t_1=9$ minutes, $W(t_1)=61.8$ °F, $t_2=15$ minutes, and $W(t_2)=67.9$ °F.
Plug in values: Plug in the values: $(67.9\,\text{°F} - 61.8\,\text{°F}) / (15\,\text{minutes} - 9\,\text{minutes})$ .
Perform subtraction: Perform the subtraction: $67.9^\circ\text{F} - 61.8^\circ\text{F} = 6.1^\circ\text{F}$ and $15$ minutes $- 9$ minutes $= 6$ minutes.
Divide differences: Divide the differences: $6.1\,^\circ\mathrm{F} / 6\,\text{minutes} = 1.0167\,^\circ\mathrm{F}$ per minute.
Conclusion: $W'(12)$ is approximately $1.0167^\circ\text{F}$ per minute. This means that at around $12$ minutes, the temperature of the water is increasing at a rate of about $1.0167$ degrees Fahrenheit per minute.

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