26 UNIT 1 • Functions, Equations, and Systems

Think About This Situation

Suppose that you were to run some experiments by changing the batteries and bulbs in a working flashlight circuit. Think about what would happen to the current flow and thus the light bulb.a How would you expect the current to change if the original battery was replaced by one

of higher voltage? What if the new battery had lower voltage?

b How would you expect the current to change if the original bulb was replaced by one of higher resistance? What if the new bulb had lower resistance?

c How would you expect the current to change if the original battery was replaced by one of higher voltage and the bulb by one of lower resistance? What if the voltage decreased and the resistance increased?

d How would you expect the current to change if both the voltage and resistance were increased? What if both the voltage and the resistance were decreased?

In this lesson, you will develop your skill in reasoning about multivariable relationships and functions.

IInvest invest iggationation 11 Combining Direct and Combining Direct and Inverse VariationInverse Variation

The flashlights we use for various lighting tasks come with many different combinations of batteries and bulbs. The relationship between current I, voltage V, and resistance R is important. If the current is too great, the bulb may break; if the current is too low, the bulb may barely light. As you work on the problems of this investigation, look for an answer to these questions:

What symbolic rules represent the relationship of current, voltage, and resistance in a simple electrical circuit?

How can relationships among several variables be written in useful equivalent forms?

1 Based on your reasoning about how changes in voltage and resistance affect current in a flashlight circuit, which of the following formulas might possibly express the numerical relationship among those variables? Be prepared to explain your reasoning.

I = V - R I = V + R I = V · R I = V _ R I = R _ V

Name:______________________________Unit 1

Lesson 2Investigation 1

LESSON 2 • Multivariable Functions 27

2 Engineers designing electrical circuits have tools to measure voltage, current, and resistance of designs they want to test. You could use those instruments to run some experiments and collect data on combinations of volts, amps, and ohms in flashlight circuits that function properly. Computer software like the “Light it Up!” custom tool can be used to simulate experiments with batteries and bulbs in search of the relationship among I, V, and R.

a. Use the simulation to find at least 5 combinations of values for these variables that will produce working battery and bulb circuits.

b. Write a formula showing how I depends on V and R in all cases. Compare your formula with those of your classmates. Resolve any differences.

c. Use the language of direct and inverse variation to explain in words how current varies as voltage and resistance change.

3 Design of a working battery and bulb circuit requires finding values for 3 related variables. In some cases, you might know values of V and R and need to calculate the corresponding value of I. In other cases, you might know values of I and R or values of I and V.

a. What is the resistance in a circuit that includes a 9-volt battery and has current flow of 2 amps? What if the circuit includes a 12-volt battery and has current flow of 5 amps? What formula expresses R as a function of V and I?

b. What size battery (in volts) is required to produce a current flow of 2.5 amps when the resistance of the circuit is 4 ohms? What if the circuit must have a current flow of 4 amps and the resistance is 7 ohms? What formula expresses V as a function of R and I?

c. How do Ohm’s Law I = V _ R and the natural connection between

multiplication and division provide algebraic justification for the formula in Part b that expresses V in terms of I and R?

4 The questions of Problems 2 and 3 asked you to write Ohm’s Law in

three different ways. You can use the formula I = V _ R to reason about

how and why changes in R and V will cause changes in I. For example,

If V stays the same and R is increased, then I will decrease because when the denominator of a positive fraction increases and the numerator stays the same, the value of the fraction will decrease.

CPMP-Tools

28 UNIT 1 • Functions, Equations, and Systems

Use the formula and similar reasoning to complete each of the following sentences about how the three variables are related.

a. If R stays the same and V is increased, then I _____________ because _____________.

b. If R and V are each doubled, then I ______________ because _____________.

c. If R is cut in half and V is doubled, then I ____________ because _____________.

5 Revisiting Roll Time, Ramp Length, and Platform Height Scientists can use theories about the effects of gravity on falling objects to deduce the relationship of roll time T, ramp length L, and platform height H, for an experiment like the one you did in Lesson 1. Ignoring possible effects of friction, theory predicts that

these variables will be related by the function T = L _

4 √ " H .

PlatformHeight

Ramp Length

a. Use the language of direct and inverse variation to describe the theoretical relationship of T to L and H. What is the constant of proportionality?

b. Use the function T = L _

4 √ " H to complete the following tables of

predicted roll time values for various combinations of ramp length and platform height. Then plot the sample (L, T) and (H, T) values on separate graphs.

Ramp Length (in feet) 4 5 6 7 8Roll Time (in sec) at 0.25-ft Height

Platform Height (in feet) 0.25 0.5 0.75 1.0 1.25Roll Time (in sec) for 8-ft Ramp

c. Compare patterns in the tables and plots in Part b to your experimental results from Lesson 1. Explain any differences.