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Alg2H
8-6 Functions of More Than One Independent Variable Lesson WK D
Combined Variation Functions and Composite Functions
Beam Strength Problem
The safe load for a horizontal beam, such as those that hold up the floor in a house, varies
directly as the breadth, directly as the square of the depth, and inversely as the length between
supports.
depth
length
| |
breadth
length
depth
|
|
breadth
a. Write a general equation expressing the safe load (s) in terms of breadth (b), depth (d), and
length (L).
b. A beam 2” by 12” by 10 feet is turned on “edge” so that the breadth is 2” and its depth is 12”.
It can support a load of 1500 lbs. Write the particular equation for this function.
NOTE: In combined variation functions, different units can be used for each variable as long as the units are
added to the definition of the variables:
Safe load = s ( lbs)
Breadth = b (in)
Depth = d (in)
Length = L (ft)
c. If the beam is laid “flat” so that the breadth is 12” and the depth is 2”, how many pounds can
it support?
Electric Light Problem
There are 4 variables concerned with the operation of an electric light:
Power = p (watts)
Voltage = v (volts)
Current = c (amps) Resistance = r (ohms)
a. Write general equations for each of the following variations:
The power is directly proportional to the product of the voltage and the current: ________________
(Use k1)
The resistance varies directly with the voltage and inversely with the current: __________________
(Use k2)
b. By appropriate substitutions, derive a function expressing power as a function of resistance
(Use k3. Define it in terms of k1 and k2)
and current.
c. Express in words how the power varies with resistance and current.
d. Write the particular equation for part (b) if a 400 watt light bulb with a resistance of 100 ohms draws
a current of 2 amps.
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Weather Balloon Problem
Helium-filled balloons are sent up into the atmosphere carrying instruments that measure
weather conditions. As the balloon ascends,
i) Its volume varies directly with the absolute temperature and inversely with the pressure
of the atmosphere.
ii) The volume of the balloon is also directly proportional to the cube of its radius.
iii) Its surface area is directly proportional to the square of its radius.
a. Identify the variables and write general equations for each of the functions above.
Volume =
pressure =
surface area =
absolute temperature =
radius =
b. By appropriate substitutions, derive an equation for the surface area in terms of volume,
by using equations from (ii) and (iii).
c. By appropriate substitutions, write an equation expressing the surface area in terms of
the temperature and pressure by using equations from (b) and (i).
d. Suppose that a balloon has a surface area of 20 square meters at ground level when it is
realized. If the air pressure is 760 millimeters of mercury (mm) and the temperature is
300 degrees Kelvin, evaluate the proportionality constant and write the particular
equation. (Add the appropriate units to each of the defined variables in part (a).)
e. As the balloon rises, it expands until it bursts. If the surface can be stretched to an area of
100 square meters before the balloon bursts, will it be able to get up to 20,000 meters
where the pressure is 50 mm and the temperatures is 200 degrees Kelvin? Justify your
answer.
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Diving Board Problem
WK D
The amount a diving board bends down from its normal position varies directly with the mass of
the person standing at the edge of the board , directly with the cube of the length of the board,
inversely with its width, and inversely with the cube of its thickness.
a. Identify the variables and write a general equations expressing amount of bending in terms of
mass, length, width, and thickness.
Bending =
Mass =
Length =
Width =
Thickness =
The bending causes stress (S) in the board that is directly proportional to the product of the
thickness and the amount of bending, and inversely proportional to the length of the board.
b. Write a general equation expressing stress in terms of these variables.
c. By appropriate substitutions, write the general equation expressing stress in terms of mass,
length, width and thickness.
d. John, who is 50 kilograms, stands on the end of the diving board 4 meters long, 40
centimeters wide and 5 centimeters thick. The stress level of 60. Write the particular equation
for part c. (Add the appropriate units to each of the defined variables in part (a).)
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Answers:
−
kbd 2
1. a) s =
L
52.08 3bd 2
b) s =
L
2. a) p = k1vc
b) p = k 3 rc 2 where k 3 =
r=
k1
k2
d) p = 1rc 2
k2v
c
3. a) (i) V =
k1T
P
b) A = k 4 v
(ii) V = k 2 r 3
A = k3 r 2
⎛T ⎞3
c) A = k 5 ⎜ ⎟
⎝P⎠
k
where k 4 = 3
k2
2
k1 ml 3
wt 3
(iii)
2
2
3
⎛T ⎞3
d) A = 37.17⎜ ⎟
⎝P⎠
4. a) B =
c) 250 lbs.
2
where k 5 = k 4 k1 3
2
⎛ 200 ⎞ 3
2
2
e) A = 37.17⎜
⎟ = 93.7 m will not burst since A < 100m
⎝ 50 ⎠
b) S =
k 2 tB
l
c) S =
k 3 ml 2
wt 2
where k 3 = k1 k 2
75ml 2
d) S =
wt 2
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