Question 67939: Most three-way light bulbs have only two filaments. The first filament may be 50 watts, for example. The second filament may be 100 watts. The three levels of brightness are obtained by using the first filament(50 watts), the second filament (100 watts), or both filaments together(150 watts). Unfortunately, your eye won't see these three steps as changes that lead to equal ratios of brightness. Switching from the first to second level yielddsd a brightness ratio of 100:50, or 2:1. However, going from the second to third level yields a ratio of 150:100, or 3:2. The two ratios are not equal.
Suppose you are asked to design a tree-way lightbulb that has a 50-watt element as its first filament , and yields equal ratios for 2nd level/1stlevel and
3rd level/2nd level.
let x be the wattage of the second filament. then the brightness of the third level will be x+50, and you want the following relationship:
2ne level / 1st level = 3rd level/2nd level
or
x /50 = x+50/x
a. Determine twhat wattage (x) should be used for the 2nd filament to meet the desired conditions.
b. Check the results of part A. What is the ratio of 2nd to 1st filament wattage? What is the ratio of 3rd to 2nd? Do they match?
c. Suppose the 1st filament is only 40 watts. what would that wattage of the 2nd filament need to be?
Answer by stanbon(75887)   (Show Source):
You can put this solution on YOUR website! 2ne level / 1st level = 3rd level/2nd level
or
x /50 = x+50/x
a. Determine what wattage (x) should be used for the 2nd filament to meet the desired conditions.
x /50 = x+50/x
Cross-multiply to get:
x^2=50x+2500
x^2-50x-2500=0
x is appox 80.9 watts
b. Check the results of part A. What is the ratio of 2nd to 1st filament wattage? What is the ratio of 3rd to 2nd? Do they match?
80.9/50 = 130.9/80
6544.81 appoximately 6545
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c. Suppose the 1st filament is only 40 watts. what would that wattage of the 2nd filament need to be?
x/40 = x+40/x
x^2=40x+1600
x^2-40x-1600=0
x is approx 64.7
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Cheers,
Stan H>
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