Download Algebra II Applications of Powers Unit Plan

Algebra II Applications of Powers Unit Plan
Now that we’ve learned the properties of powers, we can put those properties to use in some real-life
applications. In this section we will be examining the following topics:
-
Scientific Notation
Solving Equations that involve Powers
Numerical Equations that Involve Powers
Falling Object Model
Pythagorean Theorem
Other Geometry Applications
I. Scientific Notation
Scientific Notation is a way of expressing the values of very large or very small quantities without filling the
board (or the universe!) with zeros. A number is in scientific notation if it is written in the form
𝑎 ∙ 10𝑏 , with certain restrictions on “a” and “b”
Definitions:
Mantissa – The name for “a.” It must be a number greater than or equal to one and strictly less than 10.
Magnitude – The name for “b.” A positive magnitude represents a large number (bigger than 10). A
negative magnitude represents a small number (smaller than 1.)
Decimal form – What we call “ordinary” numbers in contrast to scientific notation numbers.
Key Idea: Moving the decimal point left adds to the magnitude. Moving it to the right subtracts from the
magnitude.
1.) Converting a number in scientific notation to decimal (standard) form
Ex: 5.21 ∙ 107 = 52,100,000
Ex: 4.8 ∙ 10−4 = 0.0004
Leading Zero – For small numbers it is advised to put a leading zero before the decimal point,
because it is easy for the decimal point to get lost, and because this makes the number of zeroes
equal the absolute value of the magnitude.
2.) Converting a number in decimal (standard) form to scientific notation
Key Idea: Start with a hidden “10^0”
Ex: 45,000 = 45,000 ∙ 100 = 4.5 ∙ 104
HW: p. 751 #1-55 odds (evens if necessary)
Ex: 0.000083 = 0.000083 ∙ 100 = 8.3 ∙ 10−5
3.) Operations with Scientific Notation
Key Idea: Each scientific notation number must have its own set of parentheses.
Ex: What is 3.8 ∙ 1012 divided by 9.4 ∙ 10−6 ? [Answer: 4.04 ∙ 1017 ]
Ex: A cube has a side length of 9.23 ∙ 1012 parsecs. What is the volume of this cube?
[Answer: 7.86 ∙ 1038 parsecs]
What about numbers too big for your calculator?
HW: p. 299 #36-39
4.) Large and Small Number Names
Key Idea: All the powers of 10 from 0 to 3 have their own number names. After that, every third
power gets a number name.
One:
100
Ten:
101
Tenth: 10−1
Hundred: 102
Hundredth: 10−2
Thousand: 103
Thousandth: 10−3
Million: 106
Millionth: 10−6
Billion:
109
Trillion: 1012
Billionth: 10−9
Trillionth: 10−12
Ex: How would we express 52 billion in scientific notation? [Answer: 5.2 ∙ 1010 ]
Ex: How would we express fifteen hundredths in scientific notation? [Answer: 1.5 ∙ 10−1 ]
5.) Scientific Notation Word Problems
Key Idea: If you are unsure of what operation to use, imagine smaller numbers. Then, apply the
correct operation, keeping in mind that every scientific notation number gets its own set of
parentheses!
Ex: There are 50 trillion cells in the human body. There are approximately one thousand people in
a school. How many human cells are in this school? Express your answer in scientific notation.
(5.0 ∙ 1013 ) ∙ (1.0 ∙ 103 ) = 5.0 ∙ 1016
Ex: The output of the US Economy is about 17 trillion dollars per year. There are about 300 million
people in the US. What is the average economic output per person in the US? Round to the
nearest thousand dollars.
(1.7 ∙ 1013 )
≈ $57,000
(3.0 ∙ 108 )
HW: p. 300-301 #45-53
6.) Shortcuts in your calculator
Key Idea: Your (blue) calculators have a “SCI” mode that makes every answer appear in scientific
notation. To access it, press “2nd” then “DRG”. Put the cursor under “SCI” and hit enter.
HW: Scientific Notation Topic Practice
Scientific Notation Quiz
II. Solving Numerical Equations that involve Powers
1.) Perfect Squares – A perfect square is a number that is a square of a whole number. These will
become important in this section.
Ex: Name the first eleven perfect squares [Answer: 0, 1, 4, 9, 16, 25, 36, 49, 64, 81, 100 ]
2.) Roots – When the base is unknown, the opposite of power is root. The most common root is
square root. When there is no number in the crook of the root symbol, the root is assumed to be a
square root.
Key Idea: The calculator tells you there is only one output for root, but even roots have two outputs,
one positive and one negative. We can represent this with the symbol ±
Ex: Solve for x: 3𝑥 2 − 8 = 19
[ Answer: 𝑥 = ±3 ]
Ex: Solve for x: (𝑥 + 4)2 = 100
[ Answer: 𝑥 = { −6, 14} ]
3.) Simplified Root Form – Often, when the answer to a problem like those above is a decimal, we do
not write the decimal, but rather use something called simplified root form (SRF). To use simplified
root form, follow these steps:
1.) Find the largest perfect square factor of the number under the root. (Assuming the root is a
square root.)
Ex: √150 = √25 ∙ 6
2.) Re-write the expression as two separate roots (The Product Property of Powers allows us to do
this since √𝑥 = 𝑥 1/2 .)
Ex: √25 ∙ 6 = √25 ∙ √6
3.) Evaluate the root that has the perfect square under it, but not the other root.
Ex: √25 ∙ √6 = ±5√6
Technically the ± is implied by the continued presence of the √ symbol in the expression, but
tradition dictates that we write it out anyway.
Ex: Re-write √72 in simplified root form. [ Answer: ±6√2 ]
Ex: Re-write √12 in simplified root form. [ Answer: ±2√3 ]
HW: p. 258 #3-5, #18-23; p. 260 #75; p. 266 #10
4.) Solving Equations Using Simplified Root Form
Ex: Solve the equation 𝑥 2 + 5 = 13 . Put your answer in simplified root form. [Answer: 𝑥 = ±2√2 ]
Ex: Solve the equation (𝑥 − 7)2 = 48 . Put your answer in simplified root form
[ Answer: 𝑥 = 7 ± 4√3 ]
HW: p. 257 #4-6;
p. 289 #16-18;
p.258-260 #12-17; #46-68; #77-78;
p. 782 #27 - 30
p. 266 #16-21;
p. 287 #20 – 23;
5.) Solving Equations Using Higher Roots – Not all problems that involve exponents have an exponent
of “2”. Many equations must be solved using “higher” roots. When a higher root is used, the value of
the exponent that is being eliminated is put in the “crook” of the root. Otherwise, it is assumed that
we are dealing with square roots.
Key Idea: Odd roots are functions, so there is only one output. The odd root of a negative number
is negative, and the odd root of a positive number is positive.
Ex: Solve for x: 𝑥 3 = 8 [ Answer: 𝑥 = 2 ]
[Note how
−
2 would not work as a solution here because ( −2)3 = −8 ]
Ex: Solve for x: 𝑥 4 = 16 [ Answer: 𝑥 = ±2 ]
[Note how the ± sign is used here, because the exponent is even.]
Key Idea: To access the higher roots in your calculator, press the value of the root that you want,
then the “2nd” button, then the “^” button, and then the number you want under the root.
Remember, in the case of even roots, there is still a negative output that your calculator can’t see!
Ex: Solve for x: 𝑥 5 = 60 [ Answer: 𝑥 ≈ 2.27 ]
HW: p. 356 #36-43 first, then #6-8, 27-35; p. 401 #5-8; p. 784 #1-5
6.) Falling Object Model – This physics equation models the height of a falling object as a function of its
starting height and the time. There are many different ways to write the Falling Object Model, we will
choose a simple way, which is ℎ = −16𝑡 2 + 𝑠 , where ℎ is the current height, 𝑡 is time and 𝑠
Book work: p. 257 example 4; #7;
p. 259 #69-73; p. 289 #39
7. Pythagorean Theorem – The Pythagorean Theorem relates the lengths of the sides of a right
triangle. The sides that form the right angle are called the legs, and the third side, which does not touch
the right angle is called the hypotenuse. The hypotenuse is always the longest side of the triangle.
The legs are assigned the variables 𝑎 and 𝑏 , and the hypotenuse is given the variable 𝑐 . The
equation of the Theorem is 𝑎2 + 𝑏 2 = 𝑐 2 .
Ex: A right triangle has legs of three and four inches. What is its hypotenuse? [Answer: 5 inches.]
Ex: A right triangle has a leg of 5 inches and a hypotenuse of 13 inches. What is the length of the
other leg? [Answer: 12 inches.]
Ex: A right triangle is isosceles (meaning the two legs are the same length.) Its hypotenuse is 20 cm.
What is the length of each of the legs? [Answer: about 14.1 inches.]
The converse of this Theorem is also true, that is, if a triangle has three sides, and you substitute
their lengths into the formula 𝑎2 + 𝑏 2 = 𝑐 2 so that the longest side is assigned to 𝑐 , then you
have proven that this triangle is a right triangle.
Ex: A triangle has sides of 10, 13 and 16 inches. Is this triangle a right triangle? [Answer : No.]
The Pythagorean Theorem can also be used to find the distance between two points on the
coordinate plane.
Ex: What is the distance between (4, 7) and (2, 3)? [Answer: About 4.47]
When we are using the Theorem to find distance, it is often re-written as (∆𝑥)2 + (∆𝑦)2 = 𝑑2
HW (distance formula): p. 691 #1-6; p. 694 #3-11
HW: Pythagorean Theorem Topic Practice
Pythagorean Theorem Quiz?
8.) Other Geometry Applications – Roots are useful to find the unknowns in a number of geometry
formulas.
Ex: A circle has an area of 100 square inches. Approximate its radius to the nearest tenth of an inch.
[Answer: 5.6 inches.]
Ex: A sphere has a volume of 100 cubic inches. Approximate its radius to the nearest tenth of an
inch.
[Answer: 2.9 inches]
Ex: A rectangular prism has a width that is twice its height and a length that is five times its height.
The volume of this prism is 5,120 cubic inches. What are the dimensions of this prism?
[Answer: 8 by 16 by 40 inches.]
HW: p. 260 #97-99; p. 357 #84,85
AOP Worksheets & Blockbuster: AOP Test