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.

Homework: Problems 1-14

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