unit 1 > functions, systems, and...
TRANSCRIPT
!Name:!____________________________________________________!
!
Unit 1 > Functions, Systems, and Equations
Lesson 2: Multivariable Functions Practice Problems
I can write and solve two-variable linear functions where the variables are represented by direct and inverse variation.
!
Investigation Practice Problem
Options Max Possible
Points Total Points
Earned Investigation 1: Combining Direct
and Inverse Functions #1, 2, 3, 4, 5 18 points
Investigation 2: Linear Functions and Equations
#6, 7, 8, 9, 10
20 points
!
!
!
________/38 points !
!
!
!
**Check your work with Mrs. Yauk’s answer key once you have earned your points for these practice problems, and earn 5 extra credit points**
34 UNIT 1 • Functions, Equations, and Systems
On Your Own
Applications
1 For each of the following functions, write a sentence describing the relationship among the variables in the language of direct and inverse variation.
a. Earned wages E at a job are a function of hours worked h and hourly pay rate r, given by the rule E = rh.
b. When members of a sailing club pool their money to buy a boat, the cost per member c depends on the number of members n and
the cost of the boat B according to c = B _ n .
c. When a flashlight shines on a flat surface at night, the brightness B of the light on that surface is a function of the distance d from the flashlight to the surface and the lumen strength L of the
flashlight beam; the rule that relates them is B = L _ d2 .
d. If a runner can average m miles per hour for quite a long run, the distance covered d is a function of the total running time t and average speed m.
2 When a car, van, or small truck is involved in a traffic accident, the likelihood that a passenger will be fatally injured depends on many conditions. Two key variables are speed and mass of the vehicle in which the passenger is riding.
a. What general relationship would you expect between the rate of fatalities in auto accidents and the speed and mass of the vehicle in which a passenger is riding?
b. What data on highway accidents would help you develop a function relating the rate of passenger fatalities, vehicle speed, and vehicle mass?
c. If A represents the fatality rate in auto accidents, s represents vehicle speed, and m represents vehicle mass, which of the following functions would you expect to best express the relation among those variables? Explain your choice.
Function I A = 200(s + m) Function II A = 200(s - m)
Function III A = 200 s _ m Function IV A = 200sm
/8
/1
/1
/2
LESSON 2 • Multivariable Functions 35
On Your Own
3 An important consideration in construction is the weight a steel or wooden beam can hold without breaking. Some beam materials are stronger than others. But for any particular material, two important variables that influence the breaking weight are the length and thickness of the beam.
4321
a. Which beam do you think would support the greatest weight? The least weight?
b. The beams differ in length and thickness. How would you expect those two variables to affect the breaking weight of a beam?
c. Breaking weight W depends on beam length L and thickness T. What sort of rule might be used to express W as a function of L and T?
d. The table that follows shows data collected by a class that used strands of raw spaghetti spanning gaps of various lengths to investigate the breaking weight of “spaghetti bridges”:
Breaking Weight in gramsNumber of Strands
1 2 3 4
Gap Length(in inches)
2 92.5 145.1 188.1 261.6
3 47.8 109.9 128.4 185.8
4 38.6 69.9 98.5 124.7
5 29.6 43.7 79.1 95.9
6 23.8 28.3 66.5 78.4
Do the patterns of change shown in the data table support or change your thinking about the sort of rule that might be used to express W as a function of L and T?
e. Based on the data in the table above, the class developed the rule
W = 137 T _ L to express W as a function of T and L.
i. Rewrite the rule to express T as a function of L and W.
ii. Rewrite the rule to express L as a function of W and T.
4 If a variable z is directly proportional to x and inversely proportional to y, the relationship of those variables can be expressed in the form
z = kx _ y , where k is the constant of proportionality. Write similar
rules to represent the relationships described in these situations.
a. The volume V of a cylindrical container is directly proportional to its height h and the square of its radius r.
b. The force F required to lift some object with a lever is directly proportional to the mass m of the object and inversely proportional to the length L of the lever.
/2/2/1/2
/4
/3
On Your Own
36 UNIT 1 • Functions, Equations, and Systems
c. The attraction F between two objects in space varies directly with the product of their masses m1 and m2 and inversely with the square of the distance d between them.
5 A group of 13 machine shop workers in Ohio regularly pooled their money to buy lottery tickets. On July 29, 1998, they won a lump sum payment of $161.5 million.
a. How much did each of the 13 winners receive, assuming they shared the winnings equally?
b. How would the amount received by each winner change if more workers had participated in the lottery pool? If fewer workers had participated?
c. How would the amount received by each winner change if the lottery jackpot had been larger? What if the lottery jackpot had been smaller?
d. Write a rule that gives the share of the winnings S for each person as a function of the lottery jackpot L and the number of people N in the pool.
e. Rewrite the rule in Part d to express N as a function of L and S.
6 The population density of any country or region is usually measured by calculating the number of people per unit of area. For example, in 1790, the United States had a population density of 4.5 people per square mile; in 2000, that figure had risen to 79.6 people per square mile. In that time period, both the number of people and the land area of the country grew.
People per square mileby state
300.0 to 9,316.079.6 to 299.97.0 to 79.51.1 to 6.9
U. S.density
is 79.6
Source: U.S. Census Bureau
a. What are some land areas that were added to the United States after 1790?
b. What function rule shows how to calculate population density D for a region from the area A and population P of that region?
/1/2/2
/1
/2
/1/1
LESSON 2 • Multivariable Functions 37
On Your Own
c. How do increases in the population of a region affect the population density? How about decreases in the population?
d. What change in population density of a region will result if both population and land area increase? What if both decrease?
e. How would you complete the following sentence to describe the dependence of population density on both total population and land area?
“The population density of a country is (directly/inversely) related to the total population and (directly/inversely) related to the area of the country.”
7 A local radio station sponsors a T-shirt toss and floppy hat drop during home pro basketball games. The T-shirts cost the radio station $8 each, and the hats cost $12 each.
a. The promotional cost C for the radio station depends on the numbers of shirts s and hats h given away at the game. Write a rule expressing C as a function of s and h.
b. How will the radio station’s cost change as the number of shirts given away increases? How will cost change as the number of hats given away increases?
c. Suppose the radio station has budgeted $2,400 per game for giveaways. Write an equation that represents the question “How many shirts and hats can the radio station give away for $2,400?”
d. List 4 solutions to your equation from Part c. Draw a graph that shows all of the possible solutions.
e. Rewrite your equation from Part c to express s as a function of h. Explain what the slope and s-intercept of this linear function tell you about the situation.
8 Aidan is planning to build a small goldfish pond in his back yard. He plans to keep fantail goldfish, which can grow to be 4 inches, and common goldfish, which can grow to be 8 inches.
a. The fish load of a pond is measured in total inches of fish lengths.
i. Suppose Aidan wants to keep 5 fantail goldfish and 8 common goldfish. When the fish reach full size, what will the fish load of the pond be?
ii. Write an algebraic rule that shows how pond fish load L depends on the number f of full-size fantail goldfish and the number c of full-size common goldfish in the pond.
iii. How will the fish load of the pond change as the number of full-size fantail goldfish increases? How will it change as the number of full-size common goldfish increases?
/2/2/1
/1
/2
/1
/4/3
/2
/1
/2
On Your Own
38 UNIT 1 • Functions, Equations, and Systems
b. Aidan is considering a pond kit that is rated for 140 inches of goldfish. How many full-size fantail and common goldfish can be supported in this pond? Be sure to include the following in your response:
• an equation that represents the question;
• a table showing several combinations of numbers of full-size fantail and common goldfish that meet the pond rating limit;
• a graph of all possible solutions.
c. As the number of common goldfish in a pond increases, how will the number of fantail goldfish that can be supported in the pond change?
9 Find three specific solutions of the linear equation 3x - 2y = 12. Then graph the full solution set of this equation.
10 Rewrite each of the following linear equations to express y as a function of x. Determine the slope and y-intercept for the graph of the solution set of each equation.
a. 2x + y = 6 b. 8x - 5y = 20 c. -4x - 3y = 15
Connections
11 Ms. Williams gives her students a 10-point quiz every week.
a. Suppose a student has an average quiz score of 8 after 5 quizzes.
i. How many total quiz points has the student earned?
ii. If the student scores 2 points on the sixth quiz, what is the student’s average quiz score?
b. Suppose the student still has an average quiz score of 8 after 15 quizzes.
i. How many total quiz points has the student earned?
ii. If the student scores 2 points on the sixteenth quiz, what is the student’s average quiz score?
c. How does a single quiz score’s impact on the average score change as the number of quizzes increases?
12 The three key variables in racing are distance, speed, and time.
a. If a runner covers 400 meters in 50 seconds, what is the runner’s average speed? What if it takes the runner 60 seconds to cover the same distance? What formula expresses average speed s as a function of distance d and time t?
b. If a NASCAR racer covers 240 miles at an average speed of 150 miles per hour, how long will the race take? What if the average speed is 180 miles per hour? What formula expresses race time t as a function of distance d and average speed s?
/5
/4
/9