Ms. Peralta s IM3 HW 5.4. HW 5.4 Solving Quadratic Equations. Solve the following exercises. Use factoring and/or the quadratic formula.

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1 HW 5.4 HW 5.4 Solving Quadratic Equations Name: Solve the following exercises. Use factoring and/or the quadratic formula

2 HW x 2 20x + 25 =

3 HW x 2 30x + 3 = x 2 12x + 9 = 144

4 HW x 2 60x + 25 = x 2 +11x + 6 = x 2 + 9x 64 = x = 31

5 HW ( 3x + 2) = y 2 48y + 9 = x x +147 = x 2 45 = 65

6 Assignment 5.5: Word Problems Name: Part I: Solve the following problems algebraically. Follow the guidelines below when doing the word problems. Steps in Solving Word Problems with Algebra 1. Define the variables that you want to find with let statements. 2. Create equation(s) that express the information given in the problem s scenario. 3. Solve using algebraic methods. 4. Consider if your answer(s) is/are reasonable. 5. Label your solution(s) appropriately. 6. Check your answer(s) with the conditions given in the problem. 1. Two consecutive integers have a product of 72. What are the two integers? 2. Two integers have a sum of 5 and a product of six. Find the two integers. 3.Two negative integers have a sum of 10 and a product of 24. Find the two integers.

7 4. A rectangle has width that is 7 more than its length and an area of 30. Find the length and width of the rectangle. 5. An object is moving in a straight line. It initially travels at a speed of 6 meters per second, and it speeds up at a constant acceleration of 4 meters per second each second. The distance d, in meters, that this object travels is given by the equation d = 2t 2 + 6t, where t is in seconds. According to this equation, how long will it take the object to travel 108 meters? 6. The entrance to an athletic field is in the shape of a parabolic archway. The archway is modeled by the equation d = 12x x 2, where d represents the distance in feet, that the arch is above the ground for any x value. a) For what values of x will the arch be 20 feet above the ground? b) How many feet wide is the base of the arch? c) What s the maximum height of the arch above the ground?

8 7. The product of two consecutive negative even integers is 24. Find the integers. 8. You have two squares. The longer side of the big square has a length of 1 foot greater than the of the smaller square. If the combined area of the two squares is 113 feet 2, find the length of the side of the smaller square. (Define any variables that you use by using let statements and/or by labeling a diagram.) 9. As illustrated below, a frame for a picture is 2½ inches wide. The picture enclosed by the frame is 5 inches longer than it is wide. If the area of the picture itself is 300 inches 2, determine the outer dimensions of the frame.

9 Part II: Projectile Motion. Quadratic Equations are often used to find maximums and minimums for problems involving projectile motion. For example, you would use a quadratic equation to determine how many seconds would be needed for a ball to reach its maximum height when it was thrown directly upward with an initial velocity of 96 feet per second from a cliff looming 200 feet above a beach. A ball is thrown directly upward from an initial height of 200 feet with an initial velocity of 96 feet per second. After how many seconds will the ball reach its maximum height? And, what is the maximum height? To analyze our problems, we will be using a formula for a freely falling body in which we can ignore any effects of air resistance. s(t) = 16t 2 + v 0 t + s 0 s(t) represents the projectile's instantaneous height at any time t v o represents initial velocity s o represents the initial height from which the projectile is released t represents time in seconds after the projectile is released In this formula, -16 is a constant is based on the gravitational force of the earth and represents ½ g = ½(-32 ft/sec 2 ) = -16 ft/sec 2. Since g, or the acceleration due to gravity, is being measured in ft/sec 2, we must also measure s(t), v o, and s o in terms of feet and seconds. Let's begin by substituting known values for variables in the formula: s(t) = 16t 2 + v 0 t + s 0 s(t) = 16t t Since the formula represents a parabola, we must find the vertex of the parabola to find the time it takes for the ball to reach its maximum height as well as the maximum height (called the apex). height as well as the maximum height (called the apex). Using the vertex formula: t = b 2a t = 96 2( 16) t = 3seconds Substituting into the projectile motion formula we have: s(t) = 16t 2 + v 0 t + s 0 s(t) = 16(3) (3) s(t) = 344 feet Therefore, if a ball is thrown directly upward from an initial height of 200 feet with an initial velocity of 96 feet per second, after 3 seconds it will reach a maximum height of 344 feet.

10 10. An object is launched at 19.6 meters per second (m/s) from a 58.8 meter tall platform. The equation for the object's height s at time t seconds after launch is s(t) = 4.9t t , where s is in meters. When does the object strike the ground? 11. An object in launched directly upward at 64 feet per second (ft/s) from a platform 80 feet high. What will be the object's maximum height? When will it attain this height? 12. The square of a positive number increased by 4 times the number is equal to 140. Find the number. 13. Find three consecutive positive odd integers such that the product of the first and third is equal to 1 less than twice the second.

11 14. A dud missile is fired straight into the air from a military instillation. The missile s height is given by the formula: h(t) = 16t t a. How high is the missile after 4.5 seconds? b. At what time will the missile reach its maximum height? c. What is the maximum height the missile will reach? d. When will the missile be 2,500 feet above the ground? e. When will the missile be 100 feet above the ground? 15. An object is 4900 ft above the ground. The object falls, and its height is given by the quadratic function: h(t) = 16t The height of the object above the ground is in feet and the time, t, is in seconds. Determine when the object hits the ground.

12 16. The height h of an object t seconds after being released can be modeled by the equation: h(t) = -½ at 2 + vt + s where a is the acceleration due to gravity, v is the upward speed of the object upon release, and s is the starting height of the object. (If the object starts on earth, then s = 0.) At the surface of the earth, acceleration a = 32 ft / s2. A model rocket is launched from the top of a cliff that is 384 ft. high with an upward speed of 160 ft / s. a. Write a specific function that represents the height of the rocket as a function of time. b. How many seconds after the launch does the rocket reach its maximum height? c. Determine the maximum height attained by the rocket (to the nearest foot). d. How many seconds after the launch does the rocket reach the ground? 17. (Use the same general equation found in problem #16) A juggler tosses a ball into the air. The ball leaves the juggler s hand 5 feet above the ground and has an initial velocity of 31 feet per second. a. Write an equation that represents the height of the ball as a function of time. b. How long will it take the ball to reach its maximum height? c. If the juggler catches the ball when it falls back to a height of 3 feet, then how long will the ball be in the air? (A diagram may help make the problem clearer.)