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1 Section 7.2 Quadratic Applications Now that you are familiar with solving quadratic equations it is time to learn how to apply them to certain situations. That s right, it s time for word problems. The applications that most frequently use quadratic equations are from geometry, though we ll also explore some situations in physics related to falling objects and gravity. QUADRATICS IN GEOMETRY The area of a rectangle is found by the product of its length and width: Area = Length Width. We could also write it as Length Width = Area. An example of applying this area formula to algebra can be found in the diagram on the right. This diagram suggests that we know the area (48 in. 2 ) but not the length (x + 2) or width (x). We can find the values of the length and width by using the formula. Length Width = Area: Legend: x = the width x + 2 = the length (x + 2) x = 48 This is a quadratic equation, but it s not yet equal to 0. x 2 + 2x = 48 x 2 + 2x 48 = 0 (x + 8)(x 6) = 0 Distribute x and then move 48 to the other side. Factor the trinomial (if possible; if not possible, use the quadratic formula.) x + 8 = 0 or x 6 = 0 Each factor could equal 0. x = - 8 or x = 6 We get two solutions, but the width, x, could never be - 8 (the side of rectangle can t be negative), so the only answer is x = 6: The legend says that x represents the width and x + 2 represents the length. So, the width is 6 inches and the length is 8 inches. Quadratic Applications page 7.2-1

2 Within that last problem of the sides of a rectangle is a word problem waiting to happen. You may want to do a quick review of Sections 3.2 and 3.3 to refresh your memory about setting up the legend (Section 3.2), drawing a diagram and recognizing a formula and solving the equation and writing a sentence (Section 3.3). That problem on the previous page might have been written like this: Rebecca wants to make a decorative wooden frame that has an area of 48 square inches (48 in 2 ). She also wants the length to be two more inches than the width. What should the dimensions (length and width) of the rectangle be? From Sections 3.2 and 3.3 we learned these important steps to solving a word problem: (1) decide how many unknown values there are... this problem has two unknowns; (2) if there are two or more unknown values, identify a sentence of comparison in this problem we have... the length (is) two more inches than the width; (3) set up the legend; Let x = the second thing mentioned... Let x = the width (4) use the comparison for the rest of the legend... x + 2 = the length (5) draw and label a diagram (like the one shown on the previous page); you may want to set up a chart; (6) identify a formula related to the problem and the diagram... Length Width = Area; (7) solve the equation generated by the formula... (x + 2) x = 48 (8) write a sentence answering the question... The width is 6 in. and the length is 8 in. Example 1: Procedure: James wants a rectangular garden to have an area of 60 square feet. He wants the width to be 4 feet less than the length. What should be the dimensions of the rectangle? Follow the guidelines above for this new rectangle. Legend: Let x = the length x 4 = the width Formula: Length Width = Area x(x!4) = 60 x 2!4x = 60 x 2!4x 60 = 0 (x 10)(x + 6) = 0 x 10 = 0 or x + 6 = 0 x = 10 or x = - 6 x Area is 60 ft. 2 x!4 x = 10 only because the side cannot be - 6 So, the length is 10 feet and the width is 6 feet. Quadratic Applications page 7.2-2

3 Exercise 1 Follow the guidelines outlined on the previous page to solve each application situation. SHOW ALL WORK, INCLUDING THE LEGEND AND A DIAGRAM. a) Sandy s bedroom is in the shape of a rectangle that has an area of 120 square feet. The width is two feet less than the length. Find the dimensions of Sandy s bedroom. b) Desio s backyard patio is in the shape of a rectangle and has an area of 36 square yards. The length is 1 yard more than twice the width. Find the dimensions of Desio s backyard patio. Quadratic Applications page 7.2-3

4 RIGHT TRIANGLES Another common shape in geometry is the right triangle. Every right triangle has a corner angle (which measures 90 ). This corner angle is called a right angle. In a right triangle, we call the two sides that form the right angle the legs of the triangle; the third side is called the hypotenuse, as shown in the diagram. The area of a right triangle is found by considering half of a rectangle. When a rectangle is cut in half by a diagonal, a line drawn from corner to corner, two right triangles are formed. The area of each right triangle formed is half the area of the original rectangle; in other words, the formula for the area of a right triangle is 1 2 Length Width = Area (of triangle) If the triangle is found on its own (without being half of a rectangle), then the area of the right triangle is one-half the product of the legs: 1 2 leg leg = Area Example 2: Ibrahaim, a professional landscaper, is creating a garden with a series of small patios, each in the shape of a right triangle. One right triangle, which has an area of 24 square feet, is designed so that the longer leg is 2 feet more than the shorter leg. How long is each leg of the right triangle? Procedure: Write the legend, draw and label the diagram, identify the formula, and... Quadratic Applications page 7.2-4

5 Exercise 2 Solve each application situation. Show all work. a) LaTanya s lawn is in the shape of a right triangle. The area of the lawn is 20 square yards. If the shorter leg of that triangle is 3 yards less than the longer leg, what are the lengths of the legs of the triangular lawn? b) Eduardo works in the remodeling division at a modern art museum. One of his jobs is to keep the different metal sculptures painted and in good condition. One outdoor sculpture has the shape of a large right triangle. Eduardo needs to know the area of the triangle so that he can figure out how much paint it will need. He finds out that the area he needs to paint is 10 square feet. If the longer leg is 3 feet less than twice the shorter leg, what are the lengths of the two legs? Quadratic Applications page 7.2-5

6 THE PYTHAGOREAN THEOREM The Pythagorean Theorem states that, in every right triangle: leg 2 + leg 2 = hypotenuse 2. Using the measures in the diagram, at right, the Pythagorean Theorem says, a 2 + b 2 = c 2. As an example, a right triangle that has legs of length 3 inches and 4 inches will have to a hypotenuse that is 5 inches long. (This is referred to as a right triangle.) We can demonstrate the Pythagorean Theorem using these sides: = = = 25 True! The nature of a right triangle is that the hypotenuse is always the longest of the three sides in a right triangle. (The two legs will always be shorter than the hypotenuse.) Here is a simple application involving the Pythagorean Theorem. Example 3: The hypotenuse of a right triangle is 10 inches. The longer leg is 2 inches more than the shorter leg. What are the lengths of the two legs? Procedure: Write the legend, draw and label the diagram, identify the formula, and... Quadratic Applications page 7.2-6

7 Exercise 3 Solve each application situation. Show all work. a) Shay s garden is in the shape of a right triangle. The hypotenuse is 15 feet, and the longer leg is 3 feet more than the shorter leg. What are the lengths of the two legs? b) The longer leg of a right triangle is 7 inches more than the shorter leg, and the hypotenuse 8 inches more than the shorter leg. What are the lengths of the three sides of the triangle? Quadratic Applications page 7.2-7

8 Answers to each Exercise Section 7.2 Exercise 1: a) The dimensions of Sandy s bedroom are 10 feet wide and 12 feet long. b) The dimensions of Desio s patio are 4 yards wide and 9 yards long. Exercise 2: a) The lengths of the legs of LaTanya s triangular lawn are 8 yards and 5 yards. b) The lengths of the legs of the triangle are 4 feet and 5 feet. Exercise 3: a) The lengths of the legs of Shay s triangular garden are 9 feet and 12 feet. b) The lengths of the sides of the triangle are 5 inches, 12 inches and 13 inches. Quadratic Applications page 7.2-8

9 Section 7.2 Focus Exercises Follow the guidelines outlined in Section 7.2 to solve each application situation. SHOW ALL WORK, INCLUDING THE LEGEND AND A DIAGRAM. SHOW ALL WORK ON YOUR OWN PAPER. 1. An artist has created a book of prints of her abstract work. The book, itself, is a work of art in that each rectangular page is a different size; the artist requires that the dimensions of each page be so that the width is always 2 inches less than the length. One page has an area of 48 square inches. Find the dimensions of that page. 2. A rectangular picture window has an area of 21 square feet. The length is one foot less than twice the width. Find the dimensions of the picture window. 3. A rectangular roof has an area of 60 square yards. The width is 1 yard less than half of the length. Find the dimensions of the roof. Quadratic Applications page 7.2-9

10 4. A stained glass window is in the shape of a right triangle. The whole window has an area of 30 square feet. The smaller leg is 4 feet less than the longer leg. What are the lengths of the two legs of the triangle? 5. A public vegetable garden is divided up in different shapes for those who wish to grow vegetables there. Mitch s piece is in the shape of a right triangle and has an area of 40 square feet. The longer leg is 6 feet less than twice the shorter leg. What are the lengths of the two legs of that triangle? 6. A right triangle s longer leg is 2 inches more than twice the shorter leg, and the hypotenuse is 1 inch more than the longer leg. Find the dimensions of the right triangle. 7. A right triangle s hypotenuse is 4 feet more than three times the shorter leg, and the longer leg is 1 foot less than the hypotenuse. Find the dimensions of the right triangle. Quadratic Applications page

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