Advanced MathAdvanced Math

Section 5.1: Modeling Problem Section 5.1: Modeling Problem SituationsSituations

Mathematical Models Mathematical Models – graphs, – graphs, tables, functions, equations, or tables, functions, equations, or inequalities that describe a situationinequalities that describe a situation

ModelingModeling – using mathematical – using mathematical modelsmodels

Section 5.1: Modeling Problem Section 5.1: Modeling Problem Situations (con’t)Situations (con’t)

There are 5 ways to model a situation There are 5 ways to model a situation (equation).(equation).

1.1. Using a TableUsing a Table2.2. Using a SpreadsheetUsing a Spreadsheet3.3. Using a GraphUsing a Graph4.4. Using an EquationUsing an Equation5.5. Using a Graphing CalculatorUsing a Graphing Calculator

**We will be using tables, graphs, and equations to **We will be using tables, graphs, and equations to model equations in class**model equations in class**

Sample 1Sample 1

Model and solve this situation:Model and solve this situation:

A CD player costs $195, including A CD player costs $195, including tax. You already have $37 and can tax. You already have $37 and can save $9 a week. After how many save $9 a week. After how many weeks can you buy the player?weeks can you buy the player?

Sample 1(con’t)Sample 1(con’t)

Number of WeeksNumber of Weeks Total Saved ($) = 37 + 9 x Total Saved ($) = 37 + 9 x (No. of wks)(No. of wks)

00 3737

11 37 + 9(1) = 4637 + 9(1) = 46

22 37 + 9(2) = 5537 + 9(2) = 55

33 37 + 9(3) = 6437 + 9(3) = 64

****

****

1616 37 + 9(16) = 18137 + 9(16) = 181

1717 37 + 9(17) = 19037 + 9(17) = 190

1818 37 + 9(18) = 19937 + 9(18) = 199

Sample 1 (con’t)Sample 1 (con’t)

The problem asks for the number of weeks until you The problem asks for the number of weeks until you will have $195.will have $195.

Let Let ww = the number of weeks = the number of weeks

Then, 9Then, 9ww = the amount in dollars you can save in = the amount in dollars you can save in ww weeks, and 37 + 9weeks, and 37 + 9w w = the total amount you will have = the total amount you will have saved in saved in ww weeks. weeks.

You will have enough money to buy the compact You will have enough money to buy the compact disc player when the total amount saved equals the disc player when the total amount saved equals the amount you need to purchase the CD player.amount you need to purchase the CD player.

Sample 1 (con’t)Sample 1 (con’t)

37 + 937 + 9ww = 195 = 195-37-37 -37-37

99ww = 158 = 15899 9 9

w = 17.6w = 17.6

18 weeks18 weeks

Try this one on your own…Try this one on your own…

Model and Solve this situation:Model and Solve this situation:

Suppose you are reading a novel that Suppose you are reading a novel that has 378 pages. You are now on page has 378 pages. You are now on page 62. Starting tomorrow, you plan to 62. Starting tomorrow, you plan to read 20 pages a day. How many days read 20 pages a day. How many days will you need to finish.will you need to finish.

Number of DaysNumber of DaysTotal # of pages Total # of pages 62 + 20 x (# of 62 + 20 x (# of days)days)

00 6262

11 62 + 20(1) = 8262 + 20(1) = 82

22 62 + 20(2) = 10462 + 20(2) = 104

33 62 + 20(3) = 12262 + 20(3) = 122

****

****

1515 62 + 20(15) = 36262 + 20(15) = 362

1616 62 + 20(16) = 38262 + 20(16) = 382

1717 62 + 20(17) = 40262 + 20(17) = 402

62 + 20D = 37862 + 20D = 378- 62 -62- 62 -62

20D = 31620D = 316

20D = 31620D = 31620 2020 20

D = 15.8D = 15.8

16 Days16 Days

Sample 2Sample 2

The same model can be used to The same model can be used to describe different situations.describe different situations.

Describe a situation that could be Describe a situation that could be modeled by the equationmodeled by the equation

5 = 3 + 0.25X5 = 3 + 0.25X

For the expression 0.25X, think of a For the expression 0.25X, think of a situation that involves 25% or a situation that involves 25% or a quarter of something.quarter of something.

Sample 2 (con’t)Sample 2 (con’t)

I ran 3 miles. How I ran 3 miles. How many times (x) many times (x) around a quarter-around a quarter-mile track must I mile track must I run to complete a run to complete a 5 mile run?5 mile run?

I have sold three I have sold three dollars worth of dollars worth of cookies at the cookies at the neighborhood bake neighborhood bake sale. How many 25 sale. How many 25 cent cookies (x) do cent cookies (x) do I need to sell to I need to sell to bring my sales total bring my sales total up to five dollars?up to five dollars?

Try this one on your own…Try this one on your own…

Describe a situation that could be modeled Describe a situation that could be modeled by the equation…by the equation…

2X + 90 = 1802X + 90 = 180

The acute angles of a right triangle are The acute angles of a right triangle are congruent. What is the measure of each congruent. What is the measure of each acute angle?acute angle?

Section 5.2: Opposites and Section 5.2: Opposites and the Distributive Propertythe Distributive PropertySimplify - (a – 5)Simplify - (a – 5)

Step One: Rewrite problem using a 1Step One: Rewrite problem using a 1

-1 (a – 5)-1 (a – 5)

Step Two: Distribute the -1Step Two: Distribute the -1

-1 (a) – (-1)(5)-1 (a) – (-1)(5)

Step Three: SimplifyStep Three: Simplify

-a – (-5)-a – (-5)

Step Four: Subtracting a Negative is the same as Step Four: Subtracting a Negative is the same as adding its opposite.adding its opposite.

-a + 5-a + 5

Sample 2Sample 2

Simplify 5m – (3m –n)Simplify 5m – (3m –n)

5m – 1 (3m-n)5m – 1 (3m-n)

5m –3m -1(-n)5m –3m -1(-n)

5m – 3m + n5m – 3m + n

2m + n2m + n

Try these on your own…Try these on your own…

Simplify: - (c + 6)Simplify: - (c + 6)

-c - 6-c - 6

Simplify: 26 - 3(h + 5)Simplify: 26 - 3(h + 5)

-3h + 11-3h + 11

Sample 3 …Sample 3 …Multistep EquationsMultistep Equations

The temperature at The temperature at noon today was 6 noon today was 6 degrees. At 9 p.m., the degrees. At 9 p.m., the temperature was -2 temperature was -2 degrees. How many degrees. How many degrees did the degrees did the temperature drop temperature drop today?today?

Let x = the number of Let x = the number of degreesdegrees

6 – x = - 26 – x = - 2-6 -6-6 -6

-x = (-2 -6)-x = (-2 -6)-x = -8-x = -8

-1x = -8-1x = -8

-1x = -8-1x = -8-1 -1-1 -1

X = 8X = 8

Sample 4 … Sample 4 … Multistep EquationsMultistep Equations

Solve the equation: 12 + 6x – 8x = 20Solve the equation: 12 + 6x – 8x = 20

12 – 2x = 20 OR 12 + (-2x) = 2012 – 2x = 20 OR 12 + (-2x) = 2012-2x = 2012-2x = 20 -12 -12-12 -12-2x = 8-2x = 8-2 -2-2 -2

x = -4x = -4

Try these on your own…Try these on your own…

The sea-level elevation The sea-level elevation for the entrance to a for the entrance to a cave was -50 m. An cave was -50 m. An exploration team exploration team reported 3 hours after reported 3 hours after entering the cave that entering the cave that they had descended to a they had descended to a level of -130 m. They did level of -130 m. They did this in 2 equal stages. this in 2 equal stages. How many meters did How many meters did they descend in each they descend in each stage?stage?

-50 – 2d = 130-50 – 2d = 130

D=40D=40

Solve the equation:Solve the equation:-7x + 18 + 2x = -7x + 18 + 2x =

3333

X = -3X = -3

Sample 5Sample 5

The sum of the The sum of the measures of the measures of the angles of a convex angles of a convex polygon with N sides polygon with N sides is 180(n-2). Can the is 180(n-2). Can the sum of the measures sum of the measures of the angles of a of the angles of a convex polygon be convex polygon be 450 degrees?450 degrees?

180 (N-2) = 450180 (N-2) = 450

180N – 360 = 450180N – 360 = 450

180N = 810180N = 810

N = 4.5N = 4.5

No, the sum of the No, the sum of the measures of the angles measures of the angles of a convex polygon of a convex polygon cannot be 450 degrees.cannot be 450 degrees.

Section 5.3: Variables on Both Section 5.3: Variables on Both SidesSides

Variable Terms Variable Terms – Terms of an expression – Terms of an expression that contain a variablethat contain a variable

Solving Equations with Variables on Solving Equations with Variables on Both SidesBoth Sides

Solve:Solve:4x = 2x + 64x = 2x + 6

Step One: Get all the variables on Step One: Get all the variables on one sideone side

4x = 2x + 64x = 2x + 6-2x -2x-2x -2x2x = 62x = 6

Step Two: Solve Step Two: Solve

2x = 62x = 62 22 2

x = 3 x = 3

Solve: Solve: 3x = x + 163x = x + 16

3x = x + 163x = x + 16-x -x-x -x

2x = 162x = 16

2x = 162x = 16 2 22 2

X = 8X = 8

Section 5.4: Inequalities with One Section 5.4: Inequalities with One VariableVariable

When you multiply or divide by a negative When you multiply or divide by a negative number, the inequality sign flips.number, the inequality sign flips.

Open Circle: < or >Open Circle: < or >

Close Circle: Close Circle: or

Sample 1Sample 1

Solve and graph the Solve and graph the inequality x + 2 < -1.inequality x + 2 < -1.

x + 2 < -1x + 2 < -1

-2 -2-2 -2

x < -3x < -3

Try this one on your Try this one on your own…own… x – 5 > -1x – 5 > -1

Sample 2Sample 2

7. 5-2u inequality graph the and Solve

40.8x- inequality graph the and Solve

Try these on your own…Try these on your own…

10. 2r - 8 inequality graph the and Solve

-6.3x - inequality graph the and Solve

Sample 3Sample 370.1.5)4(x inequality graph the and Solve

Try this one on your own…Try this one on your own…

5).--3(x36 inequality graph the and Solve

Sample 4: Modeling Situations with Sample 4: Modeling Situations with InequalitiesInequalities

Jamal can afford to Jamal can afford to spend $50 to buy spend $50 to buy some concert tickets some concert tickets and pay for parking. and pay for parking. Model this situation.Model this situation. Tickets: $18.00Tickets: $18.00 Parking: $5.00Parking: $5.00

Try this one on your Try this one on your own…own… Katerina has $123 in her Katerina has $123 in her

checking account. With checking account. With the extra earnings from the extra earnings from her job, she can deposit her job, she can deposit $50 a week. When she $50 a week. When she reaches $500 in her reaches $500 in her checking account, she checking account, she plans to open a savings. plans to open a savings. Model this situation. Model this situation.

50050123 w

Sample 5: Modeling Situations with Sample 5: Modeling Situations with InequalitiesInequalities

Jamal borrowed $50 Jamal borrowed $50 from his mom for the from his mom for the concert . He repays concert . He repays the loan at the rate of the loan at the rate of $6 per week. When $6 per week. When will his debt be under will his debt be under $20?$20?

Try this one on your Try this one on your own..own.. Yuri has 17 minutes of Yuri has 17 minutes of

music on a 90-minute music on a 90-minute cassette. How many 5-cassette. How many 5-minute songs can he minute songs can he still get onto the still get onto the cassette?cassette?

songs 145

314

90517

s

s

Section 5.5: Rewriting Equations and Formulas

Solve D = rt for t.

Try this one on your own… Solve A = 1/2bh for b

Sample 2

Solve 2x + y = 180 for y.

Try this one on your own…

5f – 9c = 160 for c

Sample 3

Solve P = 2L + 2W for W

Try this one on your own… W = 3M – 4K for K

Sample 4

Suppose there is an 8% sales tax on all items purchased at a craft supplies store. Write a formula to show the total amount T,

including tax, that you would pay for items at the store that cost C dollars altogether.

Try this one on your own…

Suppose Felicia has saved $140 and plans to save an additional $15 each week out of the salary she makes at her part-time job. Write a formula to show the total amount D

that she has saved after W weeks.

Section 5.6: Using Reciprocals

Solve. 15 = 6h

(-2/3)x = 97

12 = (4/5)x + 4

Try these on your own…

14w = 49W = 3.5

(11/9)x = 165X = 135

-10 = (-4/5)y + 6Y = 20

Sample 2

Leela wants to use her new graphic calculator to see the graph of the equation 2x + 3y = 6. She pressed the (y=) key to enter the equation. Here is what she saw:

:Y1 = :Y2 =

:Y3 =:Y4 =

Rewrite the equation so that Leela can enter it on her calculator.

Try this one on your own…

Wong wants to graph the equation -3x+4y=12 on his graphing calculator. How can he rewrite the equation so that he can enter it. y = 3 + (3/4)x

Roberto and two roommates ordered take-out shrimp, and the three agreed to split the cost (c) equally. Let s = Roberto’s share of the cost. Write an equation that describes the situation.

Try this one on your own…

Carlotte Mendez overhauled her tractor and mowed two of her five equal-sized fields. Overhauling her tractor took her 1.5 hours. Let t = time it takes her to mow all five fields, and let s = the time for the work she has already done. Write an equation that describes the situation.

s = 1.5 + (2/5)t

Section 5.7: Area FormulasSection 5.7: Area Formulas

Parallelogram - Area = Base x HeightParallelogram - Area = Base x Height

Triangle – Area = ½ x Base x HeightTriangle – Area = ½ x Base x Height

Trapezoid – Trapezoid –

Area = ½ x the sum of the bases x heightArea = ½ x the sum of the bases x height

Area = ½ x (b1 + b2) x heightArea = ½ x (b1 + b2) x height

ParallelogramParallelogram

Find the area of a Find the area of a parallelogram with parallelogram with base 10 cm and area base 10 cm and area of 25 cm squared.of 25 cm squared.

A = BhA = Bh

25 = 10h25 = 10h

25 = 10h25 = 10h

10 1010 10

2.5 = h2.5 = h

TrapezoidTrapezoid

Step One: Draw a picture Step One: Draw a picture and label the partsand label the parts

Step Two: Fill in the area Step Two: Fill in the area formula.formula.

A = ½(B1 + B2)hA = ½(B1 + B2)h

15600 = ½(x + 2x)13015600 = ½(x + 2x)13015600 = ½ (3x) 13015600 = ½ (3x) 13015600 = (1.5x)13015600 = (1.5x)130

15600 = 195x15600 = 195x15600 = 195x15600 = 195x 195 195195 195

80 = x80 = x

Janelle Rose wants to buy a Janelle Rose wants to buy a trapezoid plot of land. She trapezoid plot of land. She knows that the border along knows that the border along the water is twice as long as the water is twice as long as the border along the street. the border along the street. The property is 130 feet tall The property is 130 feet tall and has a total area of and has a total area of 15600. How long is the 15600. How long is the border along the water?border along the water?

Try these on your own…

Find the area of the trapezoid whose has one base of 6cm, another of 8cm, and a height of 4.3cm.

The area is 30.1 centimeters squared.

The side face of the control tower at an airport are trapezoidal. Each side face has an area of 425 square feet. The edges along the floor each measure 18 feet. The trapezoids each have a height of 17 feet. What is the measure of each edge along the ceiling?

The edge along the ceiling each measure 32 feet.

Section 5.8: Systems of Equations Section 5.8: Systems of Equations in Geometryin Geometry

The measure of an The measure of an acute angle of a right acute angle of a right triangle is four times triangle is four times the measure of the the measure of the other acute angle. other acute angle. Find the measure of Find the measure of eacheach angle. angle.

Y = 4XY = 4X

X + Y = 90X + Y = 90

X + 4X = 90X + 4X = 905X = 905X = 90

5X=905X=905 55 5

X = 18X = 18

11STST: 18 degrees: 18 degrees22ndnd: 72 degrees: 72 degrees

Try this one on your own…Try this one on your own…The measure of one acute angle of a The measure of one acute angle of a right triangle is 12 degrees more than right triangle is 12 degrees more than the measure of the other acute angle. the measure of the other acute angle. Find the measure of each acute angle.Find the measure of each acute angle.

Y = X + 12Y = X + 12X + Y = 90X + Y = 90

X + (X + 12) = 90X + (X + 12) = 90X + X + 12 = 90X + X + 12 = 90

2X + 12 = 902X + 12 = 90

2X + 12 = 902X + 12 = 90-12 -12-12 -12

2X = 782X = 78

2X = 782X = 78 22 22

X = 39X = 39

11STST: 39 DEGREES: 39 DEGREES22NDND: 51 DEGREED: 51 DEGREED

Solutions of SystemsSolutions of Systems

Two or more equations that state Two or more equations that state relationships that must all be true at the relationships that must all be true at the same time are called same time are called system of system of equations.equations.

Example: Example: y = 4x and x + y = 90y = 4x and x + y = 90

The values of the variable that make both The values of the variable that make both equations true at the same time are the equations true at the same time are the solution of a system.solution of a system.

Example: Example: x = 18; y = 72x = 18; y = 72