ball at time t The acceleration due to gravity is g 32 feet per second per

# Ball at time t the acceleration due to gravity is g

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ball at time t . The acceleration due to gravity is g = 32 feet per second per second. If the initial velocity of the ball is v 0 = 272 feet per second, find the speed of the ball after t = 6 seconds. 48 . A ball is thrown vertically upward. Its velocity t seconds after its release is given by the formula v = v 0 - gt, where v 0 is its initial velocity, g is the acceleration due to gravity, and v is the velocity of the ball at time t . The acceleration due to gravity is g = 32 feet per second per second. If the initial velocity of the ball is v 0 = 470 feet per second, find the speed of the ball after t = 4 seconds. 49 . Even numbers. Evaluate the ex- pression 2 n for the following values: i) n = 1 ii) n = 2 iii) n = 3 iv) n = - 4 v) n = - 5 vi)Is the result always an even num-ber? Explain. ii) n = 2 iii) n = 3 iv) n = - 4 v) n = - 5 vi) Is the result always an odd number? Explain. Answers 1. - 186 3. - 24 5. 7 7. 13 9. 138 11. - 134 13. 1 15. - 72 17. - 2 19. 1 21. 69 23. 0 25. 9
186 CHAPTER 3. THE FUNDAMENTALS OF ALGEBRA 27. 36 29. - 6 31. - 71 33. 1 35. 5 37. 46 39. - 29 41. 256 feet 43. 110 degrees 45. - 409 F 47. 80 feet per second 49. i) 2 ii) 4 iii) 6 iv) - 8 v) - 10 vi) Yes, the result will always be an even number because 2 will always be a factor of the product 2 n .
3.3. SIMPLIFYING ALGEBRAIC EXPRESSIONS 187 3.3 Simplifying Algebraic Expressions Recall the commutative and associative properties of multiplication. The Commutative Property of Multiplication. If a and b are any inte- gers, then a · b = b · a, or equivalently, ab = ba. The Associative Property of Multiplication. If a , b , and c are any inte- gers, then ( a · b ) · c = a · ( b · c ) , or equivalently, ( ab ) c = a ( bc ) . The commutative property allows us to change the order of multiplication without affecting the product or answer. The associative property allows us to regroup without affecting the product or answer. You Try It! EXAMPLE 1. Simplify: 2(3 x ). Simplify: - 5(7 y ) Solution. Use the associative property to regroup, then simplify. 2(3 x ) = (2 · 3) x Regrouping with the associative property. = 6 x Simplify: 2 · 3 = 6. Answer: - 35 y The statement 2(3 x ) = 6 x is an identity . That is, the left-hand side and right-hand side of 2(3 x ) = 6 x are the same for all values of x . Although the derivation in Example 1 should be the proof of this statement, it helps the intuition to check the validity of the statement for one or two values of x . If x = 4, then 2(3 x ) = 2(3( 4 )) and 6 x = 6( 4 ) = 2(12) = 24 = 24 If x = - 5, then 2(3 x ) = 2(3( - 5 )) and 6 x = 6( - 5 ) = 2( - 15) = - 30 = - 30 The above calculations show that 2(3 x ) = 6 x for both x = 4 and x = - 5. Indeed, the statement 2(3 x ) = 6 x is true, regardless of what is substituted for x .
188 CHAPTER 3. THE FUNDAMENTALS OF ALGEBRA You Try It! EXAMPLE 2. Simplify: ( - 3 t )( - 5). Simplify: ( - 8 a )(5) Solution. In essence, we are multiplying three numbers, - 3, t , and - 5, but the grouping symbols ask us to multiply the - 3 and the t first. The associative and commutative properties allow us to change the order and regroup.