newton's law of cooling common core algebra 2 homework answers

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Newton's Law of Cooling, Revisited

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• Algebra II Module 3, Topic D , Lesson 28: Student Version
• Algebra II Module 3, Topic D, Lesson 28: Teacher Version

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Common Core Algebra II.Unit 4.Lesson 14.Newton's Law of Cooling

Middle school / math / algebra.

emathinstruction

Oct 7, 2016

In this lesson we see how to transform a simple exponential function so that it fits a cooling curve.

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Newton's Law of Cooling - Algebra

New york state common core math module 3, algebra i, lessons 23.

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Chapter 6 – Exponential and Logarithmic Functions

Topic 6.6 – Newton’s Law of Cooling

Newton’s Law of Cooling uses the standard exponential growth/decay model similar to Compounding Interest and Radioactive Decay .

Slideshow: Full – 4 per page – 9 per page

Algebra Copyright © 2022 by Mike Weimerskirch and the University of Minnesota Board of Regents is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License , except where otherwise noted.

FREE K-12 standards-aligned STEM

curriculum for educators everywhere!

Find more at TeachEngineering.org .

• TeachEngineering
• Newton's Law of Cooling

Hands-on Activity Newton's Law of Cooling

Grade Level: 7 (6-8)

Time Required: 45 minutes

The activity may be conducted by using a non-expendable (and highly reusable) BASIC Stamp microcontroller, a temperature probe and other supplies, estimated at $57, or a simple thermometer; see the Materials List for details.

Group Size: 28

Activity Dependency: None

Subject Areas: Measurement, Number and Operations, Physical Science, Physics

NGSS Performance Expectations:

TE Newsletter

Engineering connection, learning objectives, materials list, worksheets and attachments, more curriculum like this, pre-req knowledge, introduction/motivation, vocabulary/definitions, troubleshooting tips, activity scaling, user comments & tips.

Heat transfer is a broad topic used in many branches of engineering. For example, mechanical engineers who design engines—from steam locomotives to modern internal combustion engines—rely on a detailed understanding of how heat moves through all types of matter. Industrial engineers use heat transfer concepts to design climate control systems for manufacturing facilities, such as foundries or refrigerated food production facilities, which integrate temperature-sensitive human workers with extreme temperature processes. Moreover, heat transfer is so critical to biological engineering that it has spawned the specialty of "bioheat" transfer, which is the study of normal functioning of the cardiovascular system as well as inherently heated treatments such as cryo-surgery and laser-based therapies.

After this activity, students should be able to:

• Record data displayed by a temperature probe.
• Plot data points to make graphs.
• Identify a heating or cooling curve as having an exponential trend.

Educational Standards Each TeachEngineering lesson or activity is correlated to one or more K-12 science, technology, engineering or math (STEM) educational standards. All 100,000+ K-12 STEM standards covered in TeachEngineering are collected, maintained and packaged by the Achievement Standards Network (ASN) , a project of D2L (www.achievementstandards.org). In the ASN, standards are hierarchically structured: first by source; e.g. , by state; within source by type; e.g. , science or mathematics; within type by subtype, then by grade, etc .

Ngss: next generation science standards - science.

Common Core State Standards - Math

View aligned curriculum

Do you agree with this alignment? Thanks for your feedback!

International Technology and Engineering Educators Association - Technology

State standards, new york - math, new york - science.

Each student needs:

• Law of Cooling Worksheet

To share with the entire class:

• 100 ml beaker
• large heat-proof bowl, at least 6 cm deep
• BASIC Stamp activity kit - Serial + USB (Text v.3.0) (includes the HomeWork Board with a built-in BASIC Stamp 2 microcontroller, LEDs, wires, resistors, capacitors, and USB cable; item code 90005 for $40, available at parallax.com) Alternatively, instead of the BASIC Stamp 2 microcontroller and associated probe and circuitry, use an analog [mercury or other] thermometer.
• Parallax AD592 temperature probe (available at parallax.com; part #28130 for $16)
• 100 Ω resistor (available at radioshack.com; part #271-1311 for $1.19 for five resistors)
• 0.22 μF capacitor (available at digikey.com; part #445-4730-ND for $0.29)
• laptop computer with PBasic software to interface with BASIC Stamp via USB cable (obtain free download of software at https://www.parallax.com/microcontrollers/basic-stamp )
• thermometer, a known one that accurately measures room temperature, for probe calibration
• hot water, tepid water and ice water, ~1 liter each
• Thermometer Program (bs2)
• (optional) projector, to display the temperature probe readings

Students should have a practical knowledge of temperature and the flow of heat from areas of high temperature to areas of low temperature. Students should be familiar with plotting points on the Cartesian plane, as well as the significance of independent and dependent axes.

What is heat transfer? Heat transfer is the study of how heat and thermal energy are generated, used and converted. The natural laws governing these phenomena influence almost every aspect of your modern life, and are important to engineers who design all kinds of devices and facilities and systems that we depend upon.

Examples are all around us: cars propelled by internal combustion engines harnessing energy from high-temperature gases; computers that rely on fans to disperse heat built up by electronic processes; many foods that are consumed after cooking to increase the bioavailability of the nutrients; and general public health that is protected through engineered heating processes in food production and sanitation processes.

Mathematically, the shape of a cooling or heating curve is an introduction to nonlinear functions. The way temperature changes in time, which is easy to understand from interaction with the physical world, is an example of physical phenomena that cannot be described by polynomials.

In this activity, you will experience heat transfer, and then apply your knowledge to solve an engineering problem!

Before the Activity

• Gather materials and make copies of the Law of Cooling Worksheet.
• Build the circuit on the Board of Education diagramed in the schematic in Figure 1 and pictured on the prototyping board in Figure 2.
• Download the PBasic software to a computer and open the attached script, Thermometer Program (bs2) .
• Attach the Board of Education to a computer with the USB cable.
• Run the code thermometer.bs2 on the Board of Education. The display will output a reading from the temperature probe.
• Calibrate the temperature probe by using a known (reliable) thermometer and changing the constant "Kal" in thermometer.bs2. To do this, put the temperature probe and the thermometer in a cup of room-temperature water for two minutes and read the temperature from the thermometer. Compare this value with the displayed reading from the temperature probe. If the displayed probe value is larger than the known temperature, decrease the value of Kal and re-run the script. If the displayed value is smaller than the known temperature, increase Kal and re-run script. Iterate this procedure until the temperature probe displays the known temperature.

• Obtain ~3 liters of water: 1 liter at room temperature (~20 ºC), 1 liter that is "hot" (~60 ºC), and 1 liter that is "cold" (~10 ºC), and be sure to have ice on hand.

With the Students

• Present Newton's law of cooling:
• The rate of cooling of a body is proportional to the temperature difference between the body and its ambient environment.
• Draw a graph, explaining that as the temperature of the soda reaches the temperature of the refrigerator, it has less to cool, so the "slope" of the graph is less steep (see Graph 1).
• Distribute a worksheet to each student. Have students predict the response of tepid water in an ice water bath, and prepare the demonstration.

• Turn on the Basic Stamp and run Thermometer Program (bs2) . Designate a student to read aloud the temperature displayed in the debug window at 15-second intervals, either from the computer screen, or from a display, if a projector is available.
• Begin the demonstration by measuring the temperature of the tepid water with the probe. With the probe still in the water, place the beaker in the prepared ice bath and leave it there. While the student announces the temperature at 15-second intervals, have the rest of the class record the temperature data on their worksheets.
• After 5 minutes, stop the program, and have students plot temperature over time on the worksheet grid.
• Discuss the results. Does the trend match the curve predicted by Newton's law of cooling?
• Have students predict the response of tepid water in a hot water bath (see Figure 4, right). Repeat steps 5-7 with a hot water bath and record results. Do the results match the prediction?
• Have students finish answering the questions on the worksheet. Collect the worksheets for grading.
• Conclude by leading a class discussion about engineering examples of applying the principles of heat transfer in the real world, including reviewing students' answers to the worksheet questions.

ambient: Present on all sides; surrounding.

BASIC Stamp: A single-board computer that runs the Parallax PBASIC language interpreter in its microcontroller. The developer's code is stored in an EEPROM, which can also be used for data storage. The PBASIC language has easy-to-use commands for basic I/O, such as turning devices on or off, interfacing with sensors, etc. More advanced commands enable the BASIC Stamp module to interface with other integrated circuits, communicate with each other, and operate in networks. The BASIC Stamp microcontroller has prospered in hobby, lower-volume engineering projects and education due to its ease of use and a wide support base of free application resources. See parallax.com.

heat: Energy transferred from one body to another because of a temperature difference.

Newton's law of cooling: The rate of cooling of a body is proportional to the temperature difference between the body and its ambient environment.

temperature: How hot or cold something is.

Pre-Activity Assessment

Cool Experiences: Ask students for examples of where they encounter heating and cooling in their daily lives. Consider the refrigerator. If you placed a room-temperature can of soda in the refrigerator and waited for it to cool, how would you expect the temperature to change? What happens to its temperature over time?

Activity Embedded Assessment

Worksheet: Have students use the attached Law of Cooling Worksheet as they conduct the activity, gathering data, graphing the data and answering questions. Review their answers to gauge their mastery of the subject matter.

Post-Activity Assessment

Heat Transfer in the Real World: Discuss with students a real engineering experience that this experiment models. The following example is one of the worksheet questions; see the answer on the Law of Cooling Worksheet Example Answers .

• Pretend you are an industrial engineer designing a factory to make cookies. You need to design a process to add melted butter to eggs, but you do not want the hot butter to cook the eggs. Butter starts to melt and eggs start to cook around at ~95 degrees Fahrenheit, so if you add a lot of melted butter to the eggs, their temperature will rise and they will cook before they make it to the cookie dough! Can you think of a general idea for how to add a gallon of butter to a gallon of eggs without the eggs getting too hot? What kind of sensors would you need in the system to make sure the eggs do not cook from the butter?

Safety Issues

Water in proximity to electronics always requires caution.

Be aware that temperature probes tend to drift, which can lead to difficulty calibrating the probe.

• For lower grades, have students discuss Newton's law of cooling and participate in the demonstration using an analog (mercury or other) thermometer.
• For upper grades, have students build the circuit that controls the temperature probe, as well as program the BASIC Stamp using the PBasic software. From a science point of view, have students predict how they think the experiment would change if they used other fluids, such as honey, oil or alcohol, and discuss the concept of specific heat. From a mathematical standpoint, explain more about exponential functions, such as the relationship between an exponential function and its derivative.

Students act as food science engineers as they explore and apply their understanding of cooling rate and specific heat capacity by completing two separate, but interconnected, tasks. They collect and graph data to create a mathematical model that represents the cooling rate, and use an exponential ...

Giancoli, Douglas C. Physics, Principles with Applications Prentice Hall: Englewood Cliffs, NJ. 1985.

Contributors

Supporting program, acknowledgements.

This activity was developed by the Applying Mechatronics to Promote Science (AMPS) Program funded by National Science Foundation GK-12 grant no. 0741714. However, these contents do not necessarily represent the policies of the NSF, and you should not assume endorsement by the federal government.

Additional support was provided by the Central Brooklyn STEM Initiative (CBSI), funded by six philanthropic organizations.

Last modified: January 16, 2020

Trending Resource : 40+ Activities for the First Week of School

The Case of the Cooling Corpse Task

I got a chance to work through the case of the cooling corpse task at a common core workshop that I attended (OGAP) during the summer of 2013. This is a fun way to bring some creativity into the classroom and explore Newton’s Law of Cooling. I am blogging about the activities that we worked through in an effort to both make them easier to find in the future and to hopefully inspire others.

We were given the following instructions that were adapted from A Watched Cup Never Cools (Key Curriculum Press, 1999).

The following is a synopsis of the first part of an unpublished story. Your task is to solve the mystery and write the rest of the story. Your conclusion must include the mathematics of the solution. It should be written in story form and incorporate the mathematics within the dialogue and other prose as smoothly and naturally as possible. Be creative, but don’t arrest the wrong individual!

It was a dark and stormy night.  Holmes and Watson were called to the scene of the murder by Inspector Lestrade of the police.  The victim was a wealthy but cruel man.  He had many enemies. The most likely suspects are the wife, the business partner, and the butler.  Each has an equally strong motive.  Each also has an alibi.  The wife claims to have spent the entire evening at the theater across town.  She was seen leaving the theater at 10:30 p.m. and returned home at 11:00 p.m., going straight up to her bedroom.  Her return was verified by the upstairs maid.  The business partner claims to have spent the evening working on papers at the office.  His wife and household staff verified that he returned home at 10:30 p.m.  The butler was on his night off.  He claims to have been at the local pub until 10:00 p.m.  The butler returned to his quarters above the carriage house at 10:05 p.m. and did not leave.  This was verified by the other servants. The body was found in the victim’s study.  Holmes arrived at the scene at 4:30 a.m.  The room was unusually warm and stuffy.  One of the police officers went to open a window.  Holmes admonished him to delay that action until he had completed his investigation of the crime scene.  He instructed Watson to determine the temperature of the body.  This was found to be 88.0°F.  Holmes questioned the servants as to the room temperature during the evening and learned that the man had liked the room warm and that the temperature in the study was always very near the current 76°F.  Holmes asked Watson to take the temperature of the body again at the conclusion of his inspection of the scene, two hours after the first reading.  It was 85.8°F. A Watched Cup Never Cools © 1999 Key Curriculum Press 

Solving this requires the use of Newton’s Law of Cooling. The presence of the number e in the equation means that students will need familiarity with logarithms in order to solve the equation needed to crack the case.

During the Common Core Workshop, we labeled this activity as addressing these standards: F-BF.1.B F-LE.1 F-LE.5

I really wanted to do this activity this year with my Algebra 2 students, but we ran out of time!

More Activities for Teaching Logarithms

Sarah Carter teaches high school math in her hometown of Coweta, Oklahoma. She currently teaches AP Precalculus, AP Calculus AB, and Statistics. She is passionate about sharing creative and hands-on teaching ideas with math teachers around the world through her blog, Math = Love.

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Section 10 - Applications of Differential Equations: Newton's Law of Cooling

In this section, we apply the techniques and theory of solving differential equations to the problems involving Newton's Law of Cooling. This type of problem involves having a warm object in cooler surroundings such as a glass of water placed in a freezer. Newton's law of cooling is a differential equation that is used to calculate the temperature of the water as a function of time.