13.7 Quadratic Equations and Problem Solving

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1 13.7 Quadratic Equations and Problem Solving Learning Objectives: A. Solve problems that can be modeled by quadratic equations. Key Vocabulary: Pythagorean Theorem, right triangle, hypotenuse, leg, sum, difference, product, times, twice, more than, less than, square of, consecutive odd, consecutive even Consecutive Numbers 1. Consecutive Number. Consecutive Odd Number 3. Consecutive Even Number 1 st number 1 st number 1 st number nd number + 1 nd number + nd number + 3 rd number + 3 rd number rd number th number th number th number + 6 Pythagorean Theorem In any right triangle Leg a Hypotenuse c a + b = c Leg b Class notes: Eample 1. Find two consecutive odd integers whose product is 3 more than their sum.
2 Eample. The length of a rectangular garden is 5 feet more than its width. The area of the garden is 176 square feet. Find the length and the width. Eample 3. The hypotenuse of a right triangle is 6 inches more than the shorter leg. The longer leg is 3 inches more than the shorter leg. Find the lengths of all three sides.
3 13.7 Eercises Solve. 1. A rectangle has an area of 4 square inches. The width is represented by 3 and the length is +. Find the dimensions.. The length of a rectangle is 3 cm more than the width. The area is 70 cm. Find the dimensions of the rectangle. 3. The length of a proposed rectangular flower garden is 6 m more that its width. The area of the garden is 7 m. Find the dimensions of the proposed flower garden. 4. A square field had 5 m added to its length and m added to its width. The field then had an area of 130 m. Find the length of a side of the original field. 5. A rock is dropped from a 784 foot cliff. The height h of the rock after t seconds is given by the equation h = 16t How long will it take the ball to hit the ground? 6. One leg of a right triangle measures 6 m while the length of the other leg measures meters. The hypotenuse measures ( 6) m. Find the length of all three sides. 7. The longer leg of a right triangle measures two feet more than twice the length of the shorter leg. The hypotenuse measures 3 feet more than twice the shorter leg. Find the length of all three sides. 8. Find the length of a ladder leaning against a building if the top of the ladder touches the building at a height of 1 feet. Also, the length of the ladder is 4 feet more than its distance from the base of the building. 9. One leg of a right triangle is 14 inches longer than the other leg. The hypotenuse is 6 inches long. Find the length of each leg. 10. Eight more than the square of a number is the same as si times the number. Find the number. 11. Fifteen less than the square of a number is the same as twice the number. Find the numbers. 1. Seven less than 4 times the square of a number is 18. Find the number. 13. Find two consecutive positive odd integers whose product is The sum of the squares of two consecutive integers is 41. Find the integers. 15. Find two consecutive odd integers such that the square of the first added to 3 times the second is 4.
4 16. The square of a number minus twice the number is 63. Find the number. 17. The length of a rectangular garden is 5 feet more than its width. The area of the garden is 176 square feet. Find the length and the width. 18 A student dropped a ball from the top of a 64 foot building. The height of the ball after t seconds is given by the quadratic equation h = 16t How long will it take the ball to hit the ground? 19. The length of one leg of a right triangle is 7 meters less than the length of the other leg. The length of the hypotenuse is 13 m. Find the lengths of the legs. 0. The numerical difference between the area and the circumference of a circle is 8π. Find the radius of the circle. (Hint: first factor out π in your equation.)
5 Some challenging problems: Factor: y 7 y 17. Factor: 3. A bo is made from a rectangular piece of metal with length 0 inches and width 10 inches by cutting out square corners of length and folding up the sides. a. What are the limitations of the size of? Eplain. b. Write a polynomial that gives the volume of the bo. c. Factor out the greatest common factor for this epression you found in part (b). d. Find the volume of the bo when = 3 inches. e. Write a polynomial that gives the outside surface area of the bo. (Hint: consider the size of the metal sheet and how much was cut out.) f. Factor out the greatest common factor for the epression you found in part (e). g. If the outside surface area of the bo is 184 square inches, find.
6 4. Cut the squares apart. Match equivalent epressions. You should get a new 4 4 square.
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