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2.3 Solving Complex Equations
This section will involve solving equations, but the algebraic manipulations will be a little more complex. It would be beneficial at this point to review radicals (inverse operations) and how to accomplish this on the calculator.
SQUARING A NUMBER comes from the concept of a SQUARE. By definition, a square is 2-dimensional and has the same measure on each side.
2x2 = 22 3x3 = 32 4x4 = 42
4 square units 9 square units 16 square units AND SO ON!
Numbers like 4, 9, 16, 25, etc. are called “perfect square” because they represent a square area.
Thus: 92 is verbalized as “nine squared.”
The reverse (or inverse) of squaring a number is taking its “square root” using the symbol
When we use this symbol, we are given the area of a square and looking for the measure of one of its sides.
EXAMPLE:
Area of the square. Length of one side of the square.
They are REVERSE operations.
CUBING A NUMBER comes from the concept of a CUBE. By definition, a cube is 3-dimensional and has the same measure on each edge.
2x2x2 = 23 3x3x3 = 33 4x4x4 = 43
8 cubic units 27 cubic units 64 cubic units AND SO ON!
Numbers like 8, 27, 64, 125, etc. are called “perfect cubes” because they represent a the volume of a 3-dimensial rectangular solid that measures the same on each side.
Thus: 93 is verbalized as “nine cubed.”
The reverse (or inverse) of cubing a number is taking its “cube root” using the symbol
When we use this symbol, we are given the volume of a cube and looking for the measure of one of its edges.
EXAMPLE:
Volume of the cube. Length of one edge of the cube.
They are REVERSE operations.
This same pattern exists for all exponents.
You can manually find the root of a number and, if you have your heart set on it, there is a relatively boring utube video (I’m sure there is more than 1). So, google your heart out. I watched part of one recently and that’s enough to satisfy my thirst for quite some time.
It is much easier to use your calculator to:
Raise a value to a power Or Find a “root” (round to 1 decimal if necessary)Practice:
x
Square
multiply a number
times itself 2 times
Square
Root
the inverse of squaring a
number, what number times itself 2 times equals
x
Cube
multiply a number
times itself3 times
Cube Root
the inverse of cubing a
number, what number times itself 3 times equals
x
To the power
of 4
multiply a number
times itself4 times
4th root
the inverse of raising a number to
the 4th power, what
number times itself 4 times equals
x
1
2
The following values can be referred to as “perfect
squares.”Perfect Cubes Perfect 4ths????
1 1 1 1
2 4 8 16
3 9 27 81
4 16 64 256
5 25 125 625
And so on……..
Example 1:The formula for the Radius of an arch window can
be used in another interesting application.
The formula to calculate the radius (R) of a portion of a circle is
W = width of the water
H = height of the water
Find the width of the water in a 12 inch radius pipe if it is 8 inches at its deepest point.
Example 2:What is involved in determining the size of a car’s engine?
An engine’s volume or Displacement (D) is D = engine displacement measured in cubic centimetersb = bore (diameter of the cylinder) measured in centimeterss = stroke (distance that the piston travels) measured in centimetersc = number of cylinders.
Find the bore necessary for a 6-cylinder engine with a 6-in stroke with 278 cubic inches of displacement, rounded to one decimal place.
Example 3:
The moment of inertia (I) of a beam is .
Note: Moment of inertia is a measure of a beam’s effectiveness
at resisting bending based on its cross-sectional shape.
I = moment of inertia of the beam measured in inches4
b = width of the beam measured in inchesd = height of the beam measured in inches
Find the height of a beam rounded to the nearest 8th of an inch if
Example 4:
Fill in the table of values accurate to three decimal places for the electrical circuit wired in parallel, using the two primary electrical formulas:
Ohm's Law V = R • I and Watt's Power Formula P = V • I V = voltage (volts), I = current (amps), R = resistance (ohms), P = power (watts)What you need to know about parallel circuits:a. Electricity passes through one or the other resistor.b. Ω is the symbol for ohm, which is the unit of measurement for resistance R.c. The subscripts for the letters serve only to
distinguish to which resistor they belong: R1 is resistor one.
d. = Rtotal
e. V1 = V2 = Vtotal
f. I1 + I2 = Itotal
g. P1 + P2 = Ptotal
Example 5:What determines how much a beam will flex and bend when it is
used in a house or a bridge?
Moment of inertia is a measure of a beam’s effectiveness at resisting bending based on
its cross-sectional shape.
Note: Deflection is simply a measurement of the amount of bend in a beam.
The point load deflection (D) of a beam is .
D = deflection measured in inchesP = weight on the beam measured in poundsL = length of the beam measured in inchesE = elasticity of the beam measured in pounds per square inch (PSI)I = moment of inertia of the beam measured in inches4
Find the moment of inertia for a beam rounded to the nearest whole number:D = 1 inP = 3250 lbs.L = 164 inE = 1,800,000 psi
The moment of inertia (I) of a beam is .
Note: Design a beam with dimensions that will have a moment of
inertia sufficient to maintain the 1 inch deflection and 3250 pound
load in the initial problem.
Section 2.3:1. 3.5 amps2. 9.1 ft3. 61 parts4. 24 slats5. 3.432 in6. 3 in7. 127 MPH
8. 2 1”4
9. 185 in10. 4.7 kΩ11. 4,033 lbs12.
Total R1 R2
V 12 5.54 6.46I 11.54 11.54 11.54R 1.04 .48 .56P 138.48 63.93 74.55
13.
Total R1 R2
V 24 24 24I 7.084 3.75 3.333R 3.388 6.4 7.2P 170.016 90 79.992
14.
Total R1 R2 R3
V 9 5.92 3.08 3.08I 11.39 11.39 4.53 7R .79 .52 .68 .44P 102.51 67.43 13.95 21.56
*** if you round to 2 decimal places as you go
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