FUNCTIONS p. 1

8.4 Model situations and relationships using a variety of basic functions (linear, quadratic, logarithmic, exponential, and reciprocal) and piecewise-defined functions.

High School / Community College / EWU

a. Choose a function suitable for modeling a real world situation presented using words or data.

Core 1
Core 2
What rule would relate time and diver’s height above the water if the tower were 20 meters above the pool and travel a distance of meters in t seconds?
The intensity of sound can be measured by several different scales. One that uses units of power is watts per square meter. It is a measure of the pressure that a sound forces on your ear.
The intensity of sound from a stereo system is a function of the listener’s distance from the speakers. Displayed in the table below are some measurements taken at various distances from speakers of a particular stereo system.
Sound Intensity table omitted due to space restrictions
·  Describe the overall pattern relating distance D and intensity I in these data.
·  Make a scatterplot of the (D,I) data pairs, and explain how the shape of the graph matches the pattern in the data table.
·  Which of he following movements will cause the greater decrease in sound intensity? Moving from 1 meter to 2 meters away from the speakers. Or Moving from 4 meters to 5 meters away from the speakers.
·  Experiment with various rules in the functions list of your graphing calculator or computer software to find a good model of the relations between I and
·  Express intensity as a function of distance. If necessary, look back at Activity 5 about light intensity for some clues.
Core 3
Integrated 3
Algebra 2, CPM
In 1912, Japan gave the United States several thousand flowering cherry trees as a symbol of friendship. Similarly, the nation of Cameroon plans to give flowering Satta trees to other countries this year. When asked how to decide which Satta trees make good gifts, Cameroon’s chief arborist explained, “We plant Satta trees when they are 6cm tall and they grow 9cm every year. The trees only flower when they are taller than 150cm.”
It is very important that the trees Cameroon gives flower this year. It would be considered an insult to give a tree that did not bloom. Luckily Cameroon has many groves of Satta trees from which to select its gifts. How old must the trees be so that they will flower within a year?
a) Discuss with your study team whether an inequality or an equation is appropriate for this situation. Be prepared to share your reasoning.
b) Write and solve a mathematical sentence to determine how old the trees can be so they flower this year.
c) Later, the arborist added, “I almost forgot to tell you! When the trees become very old, they stop flowering. Make sure you choose trees that are no more than 240cm tall.”
Discuss with your team how you can use this additional information to make sure you choose trees that will flower. Be prepared to share your answer with the class.
The Alpine Music Club is going on its annual music trip. The members of the club are yodelers and they like to play the xylophone. This year they are taking their xylophones on a gondola to give a performance on the top of Mount Monch. The gondola conductor charges $2 for each yodeler and $1 for each xylophone. It costs $40 for the entire club including the xylophones to ride the gondola. Two yodelers can share a xylophone so the number of yodelers on the gondola is twice the number of xylophones. How many yodelers and how many xylophones are on the gondola?
Algebra 2
It is not in our curriculum
Algebra 2/Trigonometry
Precalculus
Currently a junior, I.M. Smart has completed 30 courses at Ferris and now has a cumulative GPA of 3.35. How many additional courses will I.M. have to get A’s in to raise his GPA to some desired value? Define a function which determines the GPA g, when given the number of additional courses taken, c. Define your variables in your own words.
A square of side length is cut from each corner of a 10 inch by 14 inch piece of cardboard and the sides folded up to make an open topped box. Write the volume of the box as a function of . What is the implied domain of this function? What is the relevant domain for the problem situation? What dimensions of the box will give the maximum volume?
CBS Television is ready to produce the next season of Survivor, aptly named Survivor Spokane. The first part of the competition involves all 30 participants surviving in a math classroom for an entire semester. Unfortunately, one of the 30 participants arrived with the highly contagious disease pre-calcula dementia. In fact, by the third day, there were already a total of 6 people infected. Find a logistical model for the spread of the disease. Be sure to define your variables. / SFCC – Beginning & Intermediate Algebra
A cell phone costs 50$ a month plus a one time activation fee of 35$. How much will this plan cost for 2 years?
SCC – Intermediate Algebra
It is in our curriculum, but I skip it / Basic Algebra
An average person burns about 2500 calories per day doing normal activities, and 3.7 calories per minute walking at a moderate pace. Write a function rule that would determine the total number of calories burned per day if a person does all of her normal activities, and walks w minutes during the day.
Intermediate Algebra
Brett, a tile setter, creates a granite design for a customers foyer. His overhead on the job is $49.00 and the materials cost $5.25 per square foot. He charges his customer $41.25 per square foot. Write a polynomial P(x) that represents his profit if the shape of the design is a square with side x feet
Algebra Concepts
A rectangular playground is to be fenced off and divided in two by another fence parallel to one side of the playground. Four hundred feet of fencing is used. Find the dimensions of the playground that maximize the total enclosed area. What is the maximum area?
Precalculus I
Construct a function that models the following situation: the income tax for a given state is computed at a rate of 5% for the first $20,000 earned and 10% for anything over that amount.
A rectangular box with volume 60ft3 has a square base. Find a function that models its surface area in terms of the length of one side of its base.

b. Determine and interpret the meaning of rates of change, intercepts, zeros, extrema, and trends.

Core 1
Core 2
Using the rule for the 10 meter platform and travel distance is , how would the rule change if the diving took place from a 10-meter platform and the gravity were the same as on the Moon? (Distance fallen in meters is given by )
Core 3
Integrated 3
Algebra 2, CPM
Algebra 2
It is not in our curriculum
Algebra 2/Trigonometry
Precalculus
Measure the dimensions of a coke can and find out the volume of the can in cubic centimeters. Suppose the aluminum that is used in making the top of the can costs .05 cents per square centimeter while the sides and bottom are made of aluminum that costs .02 cents per square centimeter. The soda can’s tab costs 1.5 cents. Maintaining the volume of the current coke can, create a cost function. Determine whether coke is packaged in the most cost effective can. Plot a table of values and show the most effective can for the price if you think it differs from the current can design. Should the soda company change the dimensions of its can? Why or why not? Is there anything about the design of the coke can that may affect the accuracy of the measurements? Should there be extra space in the can (how much) or should it be filled to the brim? / SFCC – Beginning & Intermediate Algebra
A graph gives a good approximation of the number of food stamp recipients in millions from 1994 to 1998. What does the slope of the graph represent?
SCC – Intermediate Algebra / Basic Algebra
Given the context above:
(a)  Graph this function.
(b)  Interpret the slope in the context of the problem.
(c)  Interpret the y-intercept in the context of the problem.
(d)  Describe the problem domain and range.
Intermediate Algebra
Interpret the meaning of the intercepts of the polynomial found in part (a)
Algebra Concepts
See above
Precalculus I
Three racers ran a race as depicted below. What is the rate of change of each, from the start to the finish? How does the rate of change of each change?
Needs units on axes; is it distance vs. time or speed vs. time or both?
According to the Theory of Relativity, the length of an object is a function of its velocity with respect to an observer.
, where is the speed of light. How does the length of the object change as the velocity increases? What does the object look like if you are traveling at the speed of light? Reword so clearer

c. Abstract mathematical models from word problems and interpret solutions in the context of these source problems.

Core 1
Core 2
Core 3
See #a above
Integrated 3
Algebra 2, CPM
Algebra 2
It is not in our curriculum
Algebra 2/Trigonometry
Precalculus
An aspirin tablet in the shape of a right circular cylinder has a height of 1/3 cm and a radius of 1/2 cm. The manufacturer also wishes to market the aspirin in capsule form. The capsule is to be 3/2 cm in total length, in the shape of a right cylinder with hemispheres attached at both ends. Find a function that represents the volume of the capsule. Find the radius of the capsule so that it’s volume is equal to that of the tablet. Include any necessary or helpful graphs complete with explanations. Then, build informative, scale models of the tablet and capsule. / SFCC – Beginning & Intermediate Algebra
SCC – Intermediate Algebra / Basic Algebra
See above
Intermediate Algebra
Interpret the solutions to , where P(x) is the polynomial from part (a).
Algebra Concepts
See above
Precalculus I
Find a formula for the area of a rectangle of side lengths and . What is the maximum area of the rectangle and when does it occur? For what values of does the answer make sense?

d. Identify and justify whether a result obtained from a function model has real world significance

Core 1
Core 2
Core 3
Integrated 3
Algebra 2, CPM
Algebra 2
Algebra 2/Trigonometry
Precalculus
The effect of the Earth’s gravity diminishes as the distance from the Earth increases. A person’s weight at a given height above sea level is described by the function where r is the Earth’s radius, h is the height above sea level, and w is the person’s weight at sea level. Construct a graph (scaled appropriately) that depicts the weight of the backpack as you climb (Increase in altitude). Be sure to include and explain all relevant and implied features of the graph.
You now work for CSI. A corpse was discovered in a motel room at midnight and its temperature was 82˚ F. The temperature of the room is kept constant at 68˚ F. Two hours later, the temperature of the corpse had dropped to 79˚ F. Given k is a constant for the object in question, S is the surrounding temperature, t represents the time and theta (of time) is the temperature at the given time, Newton’s Law of Cooling states:
Find the time of death to the nearest minute. Be sure to include a graph of hours
since death as a function of body temperature. / SFCC – Beginning & Intermediate Algebra
SCC – Intermediate Algebra / Basic Algebra
See above
Intermediate Algebra
Determine which of the solutions in part (c) have real world significance and why.
Algebra Concepts
A person standing close to the edge on top of a 200-foot building throws a baseball vertically upward. The quadratic function models the ball’s height above the ground, s(t), in feet, t seconds after it was thrown. How many seconds does it take until the ball hits the ground?
Precalculus I
The current population of lynx in Montana is estimated at 1500. If the growth rate is 10% per year what was the population
a) 10 years ago; b) 100 years ago, c) 1 million years ago. What do these numbers represent?