Models and Applications

Transcription

1 Models and Applications 1 Modeling Tis Not tis 2 In Tis Section Create mat model from verbal description Simple interest problems Percentage problems Geometry formulas Literal equations Angle measurements Uniform motion Tip practice eac of tese 3 1

2 Modeling Problem solving is te ability to use information, tools, and our own skills to acieve a goal Real Life Problem Create Mat Model Solve Equation Te process of taking a verbal description of te problem and developing it into an equation tat can be used to solve te problem is matematical modeling Te equation tat is developed is te matematical model 4 Steps for Solving Problems Step 1: Identify Wat You Are Solving For Step 2: Give Names to te Unknowns Step 3: Translate into te Language of Matematics Step 4: Solve te Equation Step 5: Ceck te Reasonableness of Your Answer Step 6: Answer te Question 5 Translating Englis Expressions Mat Symbols and te Words Tey Represent Add (+) Subtract ( ) Multiply ( ) Divide ( / ) sum difference product quotient plus minus times divided by greater tan subtracted from of per more tan less twice ratio exceeds by less tan double in excess of decreased by added to fewer increased by altogeter 6 2

3 Expression for Equals (=) is is equal to equals yields results results in te result is is te same as was 7 Translating Sentences into Expressions : Translate eac of te following into a matematical statement. a.) Twelve more tan a number is 25. x + 12 = 25 b.) One-tird of te sum of a number and four yields ( x 4) 6 8 Practice 2( x 5) =

4 Practice Translate eac prase to an algebraic expression. Let x represent te unknown number. 4 more tan an unknown number 2 less tan twice an unknown number Te sum of twice a number and 29 Twice te difference of a number and ( 3) 4 + x 2x 2 2x (x ( 3)) Te quotient of a number and 13, increased by 5 x Simple Interest I = Prt Interest (I) is money paid for te use of money Te amount borrowed is called te principal (P) Te rate of interest (r), expressed as a percent, is te amount carged for te use of te principal for a given period of time, usually on a yearly basis Time (t) is te elapsed time, usually in years 11 Jose received a bonus ceck for $1000. He invested it in a mutual fund tat earned 6.75% simple interest. Find te amount of interest Jose will earn in two years. Te principal P = $1000 Te interest rate r is 6.75% r = Te time t =2 I = Prt = (1000)(0.0675)(2)= 135 Jose will earn $135 interest at te end of two year 12 4

5 Solving an Equation Involving Percent We ave two ways to solve tese By proportions n q p 100% By translating into a mat equation tis is te preferred metod 13 A number is 9% of 65 Find te number Step 1: Identify We want to know te unknown number Step 2: Name Let n represent te number Step 3: Translate A number is 9% of 65 n = Step 4: Solve te equation Step 5: Ceck Step 6: Answer te Question n = = = is 9% of Find te unknown in eac percent question Practice Wat is 67% of 140? Wat is 5% of 30? 35% of wat number is 70? 15 is 6% of wat number? Wat percent of 50 is 36? 270 is wat percent of 150? = = / 0.35 = /.06 = / % = 72 % 270 / % = 180 % 15 5

6 Figure Square Area: A = s 2 s Perimeter: P = 4s Rectangle Area: A = lw w Perimeter: P = 2l + 2w l 1 Triangle Area: A = 2b a c Perimeter: P = a + b + c b 16 Figure Trapezoid Area: A = 2(B + b) b a c Perimeter: P = a + b + c + B B Parallelogram Area: A = b a b b Perimeter: P = 2a + 2b a Circle Area: A = r 2 r Circumference: C = 2 r = d d 1 17 Figure Cube Volume: V = s 3 Rectangular Box w s Spere s l r s Surface Area: S = 6s 2 Volume: V = lw Surface Area: S = 2lw +2l + 2w Volume: V r Surface Area: S = 4 r

7 Figure Rigt Circular Cylinder Volume: V = r 2 r Surface Area: S = 2 r r Cone r Volume: V r 19 Literal Equations A literal equation contains multiple variables We are often asked to solve for one variable : Solve for b Solution y mx b y mx b b y mx 20 Te area of a trapezoid is Solve te formula for b A ( a b) 2 A ( a b) 2 2 A ( a b) 2A a b 2A a b 21 7

8 Practice 5 C F 32 9 will convert temperature in degrees Fareneit, F, to te equivalent temperature in degrees Celsius, C Use tis formula to convert 50 O F to Celsius 5 C C 9 22 Angles Complementary angles add to 90 Eac angle is called te complement of te oter (90 x) x complementary angles Supplementary angles add to 180 Eac angle is called te supplement of te oter (180 x) x supplementary angles 23 Angle A and angle B are complementary angles, and angle A is 21º more tan twice angle B Find te bot angles Step 1: Identify We are looking for two angles wose sum is 90 B A Step 2: Name Let a represent te measure of angle A Step 3: Translate Angle A is 21 more tan twice te measure of angle B a = m b a = (90 a) 24 8

9 Angle A and angle B are complementary angles, and angle A is 21º more tan twice angle B Find te bot angles a = (90 a) Step 4: Solve te equation a = (90 a) a = a a = 201 2a 3a = 201 a = 67 b = = 23 Step 5: Ceck Step 6: Answer te Question = 90 Te two complementary angles measure 67 and Practice Find two complementary angles suc tat te measure of te second angle is 40 less tan te first Let x and x - 40be temeasure of teangles Ten x x x x 130 x 65 and x O O Teangles are 65 and Uniform Motion Objects tat move at a constant velocity (speed) are said to be in uniform motion. If an object moves at an average speed r, te distance d covered in time t is given by te formula d = rt r d t 27 9

10 Mari jogs at an average rate of 8 kilometers per our How long would it take er to jog 14 kilometers? Step 1: Identify We are looking for te lengt of time it would take Mari to jog 14 kilometers Step 2: Name Let t represent te lengt of time it would take Mari to jog 14 kilometers Step 3: Translate Organize te information in a table Distance, km Rate, kp Time, ours 14 8 t d = rt 14 = 8t 28 Mari jogs at an average rate of 8 kilometers per our How long would it take er to jog 14 kilometers? 14 8t Step 4: Solve 14 t 8 t 1.75 Step 5: Ceck d = rt Step 6: Answer te Question It takes Mari 1.75 ours (or 1 our and 45 minutes) to run 14 kilometers 14 = (8)(1.75) 14 =