Questions and Answers #8 Algebra Foundations

Questions and Answers Sheet 8 Algebra Foundations Question #1 Andreas buys a new sailboat for $15,540. He estimates that the boat will depreciate by 5% each year. Which exponential function models this situation? F. 𝑦 = 15,540(1.05)𝑥 G. 𝑦 = 15,540(0.95)𝑥 H. 𝑦 = −15,540(0.95) 𝑥 J. 𝑦 = −15,540(1.05)𝑥 Answer: The exponential decay model is given by, 𝑦 = 𝑎(1 − 𝑟)𝑡 where, a is the initial amount r is the decay rate t is the time period (1-r) is the decay factor From the given data the following equation is formulated. 𝑦 = 15,540(1 − 0.05) 𝑥 𝑦 = 15,540(0.95) 𝑥 Therefore, the correct option is G. Question #2 (2𝑦 − 7)(2𝑦 + 7) Answer: Considering the question, we have: ⇒ (2𝑦 − 7)(2𝑦 + 7) = (2𝑦)(2𝑦) + (2𝑦)(7) − (7)(2𝑦) − (7)(7) ⇒ (2𝑦 − 7)(2𝑦 + 7) = 4𝑦 2 + 14𝑦 − 14𝑦 − 49 Answer ⇒ (2𝑦 − 7)(2𝑦 + 7) = 4𝑦 2 − 49 Question #3 If A is 3 × 3 with rank A = 2 , show that the dimension of the null space of A is l. Answer: To solve this question we shall use rank nullity theorem. Rank A+nullity A= no. of columns in A Solution: Dimension of null space of A=nullity of A no. of columns of A=3 Rank A=2 nullity A= no. of columns in A-rank A =3-2 =1 nullity A=1 dimension of null space =1 Question #4 Multiply the algebraic expressions using the FOIL method, and simplify. (2t+3)(t-1) Answer: Step 1 The given expression is (2t + 3)(t - 1) FOIL method: (a+b)(c+d)=ac+ad+bc+bd Step 2 (2𝑡 + 3)(𝑡 − 1) = 2𝑡 ⋅ 𝑡 + 2𝑡(−1) + 3𝑡 + 3(−1) = 2𝑡 2 − 2𝑡 + 3𝑡 − 3 = 2𝑡 2 + 𝑡 − 3 Question #5 Factor each of the algebraic expressions completely. 12𝑥 3 − 52𝑥 2 − 40𝑥 Answer: We find factor of 12𝑥 3 − 52𝑥 2 − 40𝑥 We find factor of 12𝑥 3 − 52𝑥 2 − 40𝑥 Now 12𝑥 3 − 52𝑥 2 − 40𝑥 = 𝑥(12𝑥 2 − 52𝑥 − 40) = 4𝑥(3𝑥 2 − 13𝑥 − 10) = 4𝑥(3𝑥 2 − 15𝑥 + 2𝑥 − 10) = 4𝑥(3𝑥(𝑥 − 5) + 2(𝑥 − 5)) = 4𝑥(𝑥 − 5)(3𝑥 + 2) Hence 12𝑥 3 − 52𝑥 2 − 40𝑥 = 4𝑥(𝑥 − 5)(3𝑥 + 2) Question #6 Multiply the algebraic expression using a Special Product Formula, and simplify. (5-y)(5+y) Answer: To simplify: (5-y)(5+y) Solution: (5-y)(5+y) On simplifying further, we get: (5 − 𝑦)(5 + 𝑦) − 52 − 𝑦 2 (using, (𝑎^{2} − 𝑏^{2}) = (𝑎 + 𝑏)(𝑎 − 𝑏)) = 25 − 𝑦 2 ⇒ (5 − 𝑦)(5 + 𝑦) = 25 − 𝑦 2 Answer: Required simplified expression is: 25 − 𝑦 2 Question #7 3 Which expression is equal to 9 × ? 2 Answer: Step 1 Expression is given by 9 × 3 2 Step 2 We can simplify the expression as following: 3 9×3 9× = 2 2 3 27 ⇒9× = 2 2 Question #8 𝐸 4𝑥 = 12 algebraically for d rounded to the nearest hundredth 1.00 1.10 0.62 0.48 Answer: Step 1 The given equation is 𝑒 4𝑥 = 12 and we have to solve it to find the value of x Step 2 To solve this question, we can take natural log both sides ln(𝑒 4𝑥 ) = ln(12) on the left hand side exponential gets cancelled we get 4x=2.48 [ the value of ln(12) = 2.48 x=4/2.48 x=0.62 so the third option is correct Question #9 Simplify the algebraic expressions by combining similar terms. 5x-8x+x Answer: Step 1 To simplify the expression: 5x-8x+x Solution: Similar terms are terms having same variables and powers. To simplify the algebraic terms, we add or subtract the similar terms. Simplifying the given expression. 5x-8x+x=5x+x-8x =6x-8x =-2x Therefore, after simplification we get -2x. Step 2 Hence, required answer is -2x. Question #10 Evaluate log 2 185 to four decimal places. Show algebraic processes. Answer: Let log 2 1 85 = 𝑥 Formula used: log 𝑏 𝑎 = So, here log 2 1 85 = log𝑒 𝑎 log𝑒 𝑏 log𝑒 185 log𝑒 2 𝑥 = log 2 1 85 𝑥= 5.22035 0.69314 = 7.5314 Hence log 2 1 85 = 7.5314 Question #11 Translate the following verbal statement into an algebraic equation and then solve: Use x for your variable. The quotient of three more than a number and six is nine. Equation ? x=? Answer: We need to translate the given verbal statement into an algebraic expression and solve it. The quotient of three more than a number and six is nine. Let, the number be x. 𝑥+3 According to the question, the equation is =9 ⇒ 𝑥+3 6 6 =9 ⇒𝑥+3=9 ×6 ⇒ 𝑥 + 3 = 54 ⇒ 𝑥 = 51 Question #12 Write a verbal expression for each algebraic expression. 1. 23f 2. 73 Answer: - 23f In verbal it can be written as: 23 × 𝑓 −73 In verbal it can be written as: 7 cubed Question #13 Write an algebraic expression for the verbal description: The product of two consecutive odd integers, the first of which is 2n-1 Answer: Given, the product of two consecutive odd integers, the first of which is 2n-1 . Since, the consecutive odd integer will be given as n+2 . The first is 2n-1 then the next will be 2n-1+2=2n+1 The next consecutive odd integer is 2n+1 . Question #14 Find the value of x so that the slope of the line through the points (x,-3) and (4,5) is Answer: Step 1 Given information: The slope of line is 2/3. The given points are (x,-3) and (4,5). Step 2 Calculation: 𝑦 −𝑦 The slope of the line passing through two point is calculated by 𝑚 = 2 1 𝑥2 −𝑥1 Step 3 The points are (x,-3) and (4,5). 2 3 = 5−(−3) 4−𝑥 2(4-x)=3(5+3) 8-2x=24 2x=8-24 2x=-16 x=-8 Step 4 Final statement: The value of x is -8. 2 3