Lesson 10 MA 152, Section 1.4 (part 2) and 1.5In this lesson, the following application problem will be covered.- Uniform Motion Problems- Flying Object Problems- Business Problems- Work Problems1. A piece of thick cardstock, 20 inches per side, is to have four equal squares cut from its corners as shown. If the edges are then to be folded up to make an open topped box with a bottom area of 256 square inches, find the depth of the box.12020xxx20 – 2x20 – 2x2. Susan drove 420 miles on a trip. If she had increased her speed by 10 mph, the trip would have been 1 less hour long. How long was the original trip?distance rate timeoriginal tripimaginary trip3. The height of an object thrown upward with an initial velocity of 32 meet per second is given by the formula tth 32162, where t is time in seconds. How long will it take the object to reach a height of 16 feet?24. The Toronto Dominion Center is 407 feet high. A ball is projected upward from the top of the Center and its position in feet (s) above the ground after t seconds isgiven by the equation, 40775162 tts. How many seconds have elapsed when the ball reaches the ground?5. A certain model of television sells ordinarily for $980 at an Elmer's Electronics and the store averages selling 4 of this model a week. Elmer found out, that for each discount of $40 off the price, he would sell an average of 1 more TV per week. How many of this model was sold, if he had the revenue of $7400 in a week from the sells?36. A local theater company has an average attendance for its plays of 300 persons and sells tickets for $6. It estimates for each $2 increase in the ticket price, 50 fewer persons will attend. How many increases of $2 need to be made for the revenue from ticket sales to equal $2000? What is the new ticket price?7. Kristy can mow a lawn in 1 hour less time than her brother Steve. Together (with same types of mowers), they can finish the job in 5 hours. How long would it take Kristy if she worked alone? Round to the nearest tenth of an hour.49. Find the dimensions of a rectangle whose area is 240 square inches and whose perimeter is 64 inches.IMAGINARY & COMPLEX NUMBERSThe imaginary unit i is defined as the solution of the equation 12x. Therefore1 and 12 ii. Number such as 3i, 2 and ,32ii are called imaginary numbers.Ex 10: Simplify each using the imaginary unit i.121100 )20 )25 )cbaA complex number is the sum or difference of a real number (a) and an imaginary number (bi). A complex number can be written in the form bia , where a is the real partand bi is the imaginary part with 1i.The following diagram shows the relationship among these sets of numbers.5Complex Numbers34 , ,3232iii Real Numbers86.2 ,45 , ,52711Imaginary Numbers ,13 ,7 ,4 iiOperations with Complex Numbers (+, -, �):Addition/Subtraction: Add or subtract as if the numbers were binomials.Multiplication of Complex Numbers: Multiply as if the numbers were binomials.Ex 11: Add, subtract, or multiply and write answers in simplest form.( ) ( )3 24 3) (6 2 ) (3 14 )) 3 7) (14 16) ( 81 2)) (3 12) (17 49)a i ib i icd i- + + =- - + =- - - - - =+ + - - =6( ) ( ) ( ) ( )( ) ( ) ( ) ( )a bi c di a c b d ia bi c di a c b d i+ + + = + + ++ - + = - + -ibcadbdacibdbciadiacbdibciadiacdicbia)()()1( ))((222) (4 5 )(2 6 )) (4 9)( 2 36)) (4 7 )e i ifg i- - =+ - - - - =- =The numbers biabia and are called complex conjugates. Division of Complex Numbers: Multiply numerator and denominator by the conjugate of the denominator and simplify.Ex 12: Divide and simplify. Write answers in complex number form, a + bi.ia312 )3) 2 44 2) 3biici-=-+=-Powers of i:7There is a pattern in the powers of the imaginary unit, i.0 4 2 2 81 5 4 91 ( 1)( 1) 1 1 1 i i i i ii i i i i i i i i= = = - - = == = = = =1 1)1)(1( 1102462 iiiiiiiiiiiiiiiii 1134723 )(1 1Notice: Even powers of i are either 1 or -1 and odd powers of i are either i or -i.To evaluate a power of i: ni1. Determine how many groups of 4 are in n by dividing n by 4.2. The power can be written as rnii where r is the remainder after dividing.3. Simplify ri. The answer will be either 1, -1, i, or -i.Ex 13: Evaluate each power.49256) ) a ib i==If the value of the discriminant is negative, the solutions of the quadratic equation can be represented as complex numbers.Quadratic Equations with Complex Roots:Ex 14: Solve each equation. 22) 2 52 2) 03 9a x xb x x- =-- + =0432 )2 xxc8Ex 15: In electronics, the formula IRV is called Ohm's law. It gives the relationship among the voltage V (in volts), current I (in amperes), and resistance R (in ohms). Find the number of volts if the current is 3 - 2i amps and the resistance is 3 + 6i
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