7.3.2How fast does soda warm up?

Newton's Law of Cooling

7-107.

THE WARM SODA, Part One

After a calculus discussion one day, Liliana, Katrina, and Gretchen are working on practice problems together in the chilly basement of the math library at Newton University.

“Yuck! This soda is nasty!” Liliana complains.

“What’s wrong?” asks Gretchen.

“This soda is too warm! It’s been sitting out for an hour.” replies Liliana. “I would have drank it sooner but I was too wrapped up in this fun calculus problem. I didn’t think my soda would warm up this fast!”

“How fast does a soda warm up?” Gretchen inquires.

“I don’t know... let’s find out with Liliana’s soda!” says Katrina as she whips out a thermometer. She quickly determines that Liliana’s soda is approximately $48^{\circ}\text{F}$ .

Predict how long you think it will take Liliana’s soda to warm up to the room temperature ( $60^{\circ}\text{F}$ ).
An hour later, Katrina measures the temperature of the soda to be approximately $54^\circ \text{F}$ . She predicts that the soda will be room temperature after another hour. “You see, it took one hour for the temperature to rise about $6^\circ$ . Therefore, it will take another hour to rise the remaining $6^\circ$ to room temperature,” she explains. Do you agree or disagree with Katrina? Explain.
Sketch a graph to show how you believe the temperature will change over time.
If the rate of the warming is not constant, then there must be some reason why the rate will change over time. What is the rate of the temperature change dependent upon? How does the temperature of the room affect the rate of temperature change of the soda?
Let $\frac { d F } { d t }$ represent the rate the soda temperature is changing with respect to time. This rate is proportional to the difference between the temperature of the room and the temperature of the soda at any given time. Write a differential equation that represents this relationship. Let $k$ be the constant of proportionality, $R$ represent the room temperature, and let $F$ be the temperature of the soda.

7-108.

WHAT DOES THE DATA TELL US?

The soda temperature data located on the Lesson 7.3.2A Resource Page (or 7.3.2 eTool ) represents the first four hours Liliana, Katrina, and Gretchen were working in the basement of the math library. Plot the time, $t$ , and soda temperature, $F$ , on your graphing calculator. Examine the graph of the data. What happens to the temperature of the soda as time increases? Why? Click in the lower right corner of the graph to view it in full-screen mode.
Describe the rate of change of the soda temperature. Is this rate constant? Write a rate statement for this relationship.
On your resource page, locate the column labeled $\frac { \Delta F } { \Delta t }$ . How are the values of $\frac { \Delta F } { \Delta t }$ calculated? With your team make sense of these values and be prepared to justify your results.
Since $\frac { \Delta F } { \Delta t }$ can be used to approximate $\frac { d F } { d t }$ , we can obtain the constant of proportionality, $k$ , by examining the relationship between $(60 - F)$ and $\frac { \Delta F } { \Delta t }$ . Use the data in the $(60 - F)$ and $\frac { \Delta F } { \Delta t }$ columns to make a scatterplot on your calculator. Use the linear regression feature to write an equation that fits the scatterplot data. The slope of this line represents the constant of proportionality, $k$ .

7-109.

THE WARM SODA, Part Two

In problem 7-107, Gretchen asked an important question: “How fast does the soda warm up?” Gretchen is asking about a rate. Look again at the data from problem 7-108 and write a rate equation ( $\frac { d F } { d t }= \text{...}$ ).

Using the data, set up a differential equation for the rate the temperature of the soda is changing. That is, write an equation that models $\frac { d F } { d t }$ , the change in temperature over time.
Solve the differential equation to write an equation for temperature, $F$ , at time $t$ . Test your solution by graphing the equation with its time and temperature data. Was it a good fit?

7-110.

THE COOLING LAB

Obtain a Lesson Lesson 7.3.2B Resource Page. Follow the instructions provided on the resource page to gather the Cooling Lab data. Then, use the process outlined in problems 7-108 and 7-109 to write and solve the differential equation that models your data. Before you begin, be sure to obtain an accurate room temperature!

7-111.

What is the antiderivative of each of the following differential equations? Use implicit integration when necessary. Solve your equations for $y$ . Be careful about introducing the constant of integration at the appropriate time. Homework Help ✎

$\frac { d y } { d x }= 7x + 3$

$\frac { d y } { d x }= 7y + 3$

$\frac { d y } { d x }= 7y^2$

$\frac { d y } { d x } = 7$

$\frac { d y } { d x }= e^y$

7-112.

Which of the following differential equations could model a population whose rate of change is proportional to the existing population? Explain your reasoning. Is it possible that there is more than one answer? Homework Help ✎

$\frac { d P } { d t }= 3P$

$\frac { d P } { d t }= 3t$

$\frac { d P } { d t }= −3P$

$\frac { d P } { d t }= 3$

7-113.

Remember Theresa, the girl who loves tangents? Well, Theresa is at it again! She has drawn tangents to a function and then erased the function. Homework Help ✎

Trace her diagram on your paper. Using the tangents, sketch a possible function.
Is your function differentiable everywhere? Justify your answer.
Could any of the points shown be the location of a local maximum or minimum? Justify your ideas.

4 short segments, highlighted center point & first approximate end point of segment as follows: first, (negative 4, comma 3), & (negative 4.5, comma 2.5), second, (negative 1, comma 1), & (negative 1, comma 0.5), third, (negative 2, comma negative 2), & (negative 2.25, comma negative 1.5), fourth, (2, comma negative 3), & (1.5, comma negative 2.75).

7-114.

Differentiate each of the following equations. Homework Help ✎

$f(x) =\frac { \operatorname { tan } ( x ) } { e ^ { - x } }$

$y^3 - 3xy = 3^y$

$g(t) = \ln(t)\sqrt { t }$

7-115.

If the graph at right represents distance over time between a space shuttle and Earth, how would you determine the average velocity? Describe the method you would use. Homework Help ✎

First quadrant, x axis labeled time, hours, y axis labeled distance, miles, decreasing curve starting at top of y axis, dropping about half way & running about a third, leveling out, running another sixth, changing from concave up to concave down, running about another sixth, then dropping to the x axis.

7-116.

What if the graph above represents the velocity in miles per hour of the shuttle? In this case, how would you determine the average velocity? Describe the method you would use. Homework Help ✎

7-117.

Multiple Choice: What is the area of the largest rectangle that can be inscribed under the graph of $y=\cos(x)$ , over the interval $-\frac { \pi } { 2 }≤ x ≤ \frac{π}{2}$ ? 7-117 HW eTool. Homework Help ✎

$1.10$

$1.12$

$1.14$

$1.16$

$1.18$

7-118.

What is the derivative of each of the following functions? Homework Help ✎

$y = e^{\sin(2x)}$

$y = \ln(\cos(2x))$

$y = \ln(e^x)$

7-119.

Integrate. Homework Help ✎

$\int 2 \operatorname { sin } ( 2 x ) e ^ { \operatorname { cos } ( 2 x ) } d x$

$- \int ( 6 x - 2 ) ^ { 7 } d x$

$\int \frac { \operatorname { ln } ( x ) } { 2 x } d x$

7-120.

Without a calculator, determine the following limits. Homework Help ✎

$\lim\limits _ { x \rightarrow \infty } ( 4 x + 2 - \frac { 3 } { x - 2 } )$

$\lim\limits _ { x \rightarrow - \infty } \frac { 2 x ^ { 2 } + x - 21 } { 2 x ^ { 2 } + 5 x - 7 }$

$\lim\limits _ { x \rightarrow 0 + } ( \operatorname { ln } ( x ) + 18 )$

$\lim\limits _ { x \rightarrow \frac { \pi } { 3 } } \frac { \operatorname { sin } ( x ) - \frac { \sqrt { 3 } } { 2 } } { x - \frac { \pi } { 3 } }$

Compute without a calculator

7-121.

Thoroughly investigate the graph of $y = \frac { 1 } { x }\cos(x)$ for $–4 ≤ x ≤ 4$ . Identify all of the important qualities, such as where the function is increasing, decreasing, concave up, and concave down. Also identify point(s) of inflection and intercepts, and provide graphs of $f^\prime(x)$ and $f^{\prime\prime}(x)$ . Be sure to justify all statements graphically and analytically. Homework Help ✎

7-122.

Feng throws a baseball directly up in the air with an initial velocity of $16$ feet per second from a height of $4$ feet. Homework Help ✎

When will the ball hit the ground?
What is the ball’s maximum height?
What is the ball’s velocity when it hits the ground?
What is the total distance the ball travels from $t = 0$ seconds to $t = 0.75$ seconds?

7-123.

Determine the values of $a$ and $b$ so that the function defined below is continuous and differentiable. Homework Help ✎

$f ( x ) = \left\{ \begin{array} { l l } { a e ^ { x } } & { \text { for } x \leq 1 } \\ { \operatorname { ln } ( x ) + b } & { \text { for } x > 1 } \end{array} \right.$

7-124.

What are the dimensions of the right triangle with hypotenuse of length $13$ that has a maximum area? Homework Help ✎

Calculus Third Edition