Classwork 8.1. Perform the indicated operation and simplify each as much as possible. 1) 24 2) ) 54w y 11) wy 6) 5 9.

Transcription

1 - 7 - Classwork 8.1 Name Perform the indicated operation and simplify each as much as possible. 1) 4 7) ) 5 4 8) ) 5 4x 9) 9 x + 49x 4) 75w 10) w y 5) 80wy 11) ) 5 9 1) 15x 6x

2 - 74-1) ) Calculate to the nearest two decimal places: ) 14x 5x 5 18) Calculate to the nearest two decimal places: ) ) Calculate to the nearest two decimal places: ) ) Calculate to the nearest two decimal + places: () (5) 4(5)( 6)

3 Class work 8.1, 8., 8. Simplify as much as possible: Name x y x x y b y x y bc 4b c a bc a bc x yz x y

4 ( ) 0. 8 ( 4 8) ( + 5 )( 5+ ). ( 5)( 5) a 7a y b 0b

5 Classwork Name Sec 8.4 Solving Radical Equations Solve each equation using the square root property. Leave your answers in radical form simplified as much as possible. 1. y = 45. x = = 9 y 4. ( x 6) = ( w + 5) = ( x + ) + 4 = 104 p 8. 4( x + 11) = 60 = 9. 7x 9 = Solve for r : π r = 84

6 Solve each equation using the squaring property of equality x + 4 = 8 1. x + 4 = x = x x = x y = y x = 7x 17. y + 1 = y w = 5 11w 19. 5p = 9 p Solve for t : d = t

7 Classwork Name Section 8.4 Applications of Radical Equations Write all radical answers in simplified form and as a decimal rounded to two places. 1. A square has area 80 square meters. Draw the square and label its sides. Find the length of a side of the square. Write an equation for the area of the square. Solve your equation to find the solution to the problem. Use your answer to find the perimeter of the square and the length of its diagonal. Side (radical) Side (decimal) Diagonal (radical) Diagonal (decimal) Perimeter (radical) Perimeter (decimal). The diagonal of a square is 1 meters long. Draw the square and label its diagonal and sides. Find the length of a side of the square. Use the Pythagorean Theorem to write an equation. Solve your equation to find the solution to the problem. Use your answer to find the area of the square and the perimeter of the square. Side (radical) Side (decimal) Area Perimeter (radical) Perimeter (decimal)

8 . The length of a rectangle is five times its width. The area of the rectangle is 80 square meters. Draw the rectangle and label its length and width. Find the length and width of the rectangle. Write an equation for the area of the rectangle. Solve your equation to find the solution to the problem. Use your answers to find the perimeter of the rectangle and the length of its diagonal Width (radical) Width (decimal) Length (radical) Length (decimal) Perimeter (radical) Perimeter (decimal) Diagonal (radical) Diagonal (decimal) 4. The length of a rectangle is three times its width. The diagonal of the rectangle is 0 meters long. Draw the rectangle and label its length, width and diagonal. Find the length and width of the rectangle. Use the Pythagorean Theorem to write an equation. Solve your equation to find the solution to the problem. Use your answers to find the area and the perimeter of the rectangle. Width (radical) Width (decimal) Length (radical) Length (decimal) Perimeter (radical) Perimeter (decimal) Area _

9 Classwork Sections 9.1 and 9. Solving Quadratic Equations Quadratic Formula: to solve Name ± 4 ax + bx + c = 0: x = b b ac a Solve each equation. Write all radical answers in simplified form and as a decimal rounded to one place. 7 1) x = 80 ) x 9 = ) 11 8( x 5) = (x + 7) 4) x 11x + 8 = 0 4 y 6) x = ) ( 9) = 100 7) + 9m 5 = 0 m 8) ( ) x + 5 = 100 9) w 4w = 10 10) 0.08( x + 1) 0.9(x + ) =. 4

10 11) x + 5 x 7 = 8 5 1) 7y + 18y + 8 = ) 1x 4x = 0 14) ( y + 6) + (y 10) = ) 9w 15 = 0 16) x 5 = x 17) x = 1x 18) t 8 = 5 t 19) = x ) ( x + 1) + ( x 4) = 1 x

11 Classwork Section 9.4 Applications of Quadratic Equations Write all answers as a decimal rounded to two places. Name A rectangle has length five more than three times its width. The area of the rectangle is 65 square meters. Find the length, width and perimeter of the rectangle. Draw the rectangle and label its length and width. Write an equation for the area of the rectangle. Solve your equation to find the solution to this problem. Width Length Perimeter. The length of a rectangle is 7 more than twice its width. A foot border is added all the way around the rectangle. The area of the new larger rectangle is 140 square feet. Find the length, width and perimeter of the new larger rectangle. Draw the rectangle and label its length, width and the border. Write an equation for the area of the rectangle. Solve your equation to find the solution to this problem. Width Length Perimeter. Mary has 10 feet of fencing to build a rectangular pig pen. She would like the area to be 540 square feet. Find the length and width of the pen. Draw the pen. Write an equation for the area of the pen. Solve your equation to find the solution to this problem. Width Length

12 One leg of a right triangle is 5 inches longer than the other leg. The hypotenuse of the triangle is 6 inches. Find the length of the legs of the triangle. Then find the area and perimeter of the triangle. Draw the triangle and label the legs. Use the Pythagorean Theorem to write an equation. Solve the equation to find the solution to the problem. Leg a Leg b Perimeter Area 5. The length of a rectangle is 7 feet shorter than its width. It diagonal is 8 feet. Find the dimensions of the rectangle. Draw the rectangle and label its length, width and diagonal. Use the Pythagorean Theorem to write an equation. Solve your equation to solve the problem. Then find the area and the perimeter of the rectangle. Width Length Perimeter Area 6. Together Kelly and Tyler can paint the living room in eight hours. Working alone Tyler can paint the living room in three hours more than it takes Kelly working alone. How long will it take each of them, working alone, to paint the living room? Write an equation and use it solve the problem.

13 Classwork Name Section 9.4 More Quadratic Applications Write all answers as a decimal rounded to two places. 1. A rectangle has length 6 more than five times its width. The area of the rectangle is 17 square meters. Find the length, width, and perimeter of the rectangle. Draw the rectangle and label its length and width. Write an equation for the area of the rectangle. Solve your equation to find the solution to the problem. Width Length Perimeter. The length of a rectangle is 9 feet shorter than its width. Its diagonal is 7 feet. Find the dimensions of the rectangle. Then find its area and perimeter. Draw the rectangle and label its length, width and diagonal. Use the Pythagorean Theorem to write an equation involving the length, width and diagonal. Solve your equation to find the solution to this problem. Width Length Perimeter Area

14 The length of a rectangle is 5 more than three times its width. A foot border is added all the way around the rectangle. The area of the new larger rectangle is 80 square feet. Find the length, width and perimeter of the new larger rectangle. Draw the rectangle and label its length, width and the border. Write an equation for the area of the rectangle. Solve your equation to find the solution to this problem. Width Length Perimeter Area 4. Together Jon and Dylan can mow the lawn in 1 hours. (It s a very large lawn.) Working alone, Dylan can mow the lawn in four hours more than it takes Jon working alone. How long will it take each of them, working alone, to mow the lawn? Write an equation and use it to solve the problem.

15 Area and Circumference of Circles The parts of a circle are the center, radius, diameter, and circumference. The early Greeks and Babylonians, circa 1000 B.C., discovered that the circumference divided by the radius is always the same number. The Greeks used one of their letters pi (π ) for as a name for the ratio. So, according to the Greeks, C D = π. Mathematician have used various numbers as an approximation for π over the centuries. The following is a summary of some the different numbers used for π. Culture Time Used Number Used Decimal Equivalent Egyptian 1650 B.E Greeks 40 B.E < π <.149 (Archimedes) < π < 71 7 Greeks 150 A.E (Ptolemy) 10 Chinese 489 A.E Hindu 50 A.E. 6, (Aryabahata) 0, 000 French (Viete) 1579 A.E Correct to nine decimal places. Abraham Sharp 1699 A.E. π correct to 71 decimal places Rutherford 1841 A.E. π correct to 08 decimal places Army Ballistic Research Laboratories Central Intelligence Agency 1949 A.E. π correct to 07 decimal places 1984 A.E. π correct to billion decimal places For a better history of π see Introduction to the History of Mathematics by Howard Eves. According to a mathematical handbook π is correct to fifty decimal places. For our purposes, we will use either the letter π,.14, or. We will use 7 π when asked to put the answer in π form. Use.14 when the radius or diameter is given in whole or decimal form. Use when the radius or diameter is given in fraction form. 7

16 Area and Circumference of Circles The formula for finding the area of a circle is A = π r. The formula for finding the circumference of a circle is C = Dπ or C = π r. Like the formulas for the area of triangles and quadrilaterals, it is wise to memorize the two formulas. Example 1: Find the area of a circle, if the diameter is 5.80cm. Solution: Since the radius is one-half of the diameter, r=1.90. So,A=(.14)(1.9) =5.574 Or rounded to two decimal places 5.5 cm. Example : Find the Circumference of a circle, if r= 56 in. Solution: D ( ) = = and C = π = = = 117 in. 7 Example : Find C in π form, if D=15. Solution: C = Dπ = 15π Example 4: Find the Area in π form, if C=0π. Solution: Since C=Dπ and C=0π, then Dπ=0π and then D=0 and r=10. So A=π(10) =100π. Example 5: Find the Circumference in π form, if A=11π. Solution: Since A=πr and A=11π, then πr =11π. So, r =11 and r=11. Then, D= and C=π.

17 Classwork Area and Circumference of Circles Name r Circle of radius r: Area A = πr Circumference C = πr Write your answers in terms of π and as a decimal rounded to two places. 1) A circle has radius 1 meters. a) Find its circumference. b) Find its area. ) A circle has radius 15.8 meters. a) Find its circumference. b) Find its area. ) A circle has diameter 1 inches. a) Find its circumference. b) Find its area. 4) A circle has circumference 0π inches. a) Find its radius. b) Find its area. 5) A circle has circumference 4 meters. a) Find its radius. b) Find its area.

18 6) A circle has circumference 4π 5 inches a) Find its radius. b) Find its area. 7) A circle has area 5π square inches. a) Find its radius. b) Find its circumference. 8) A circle has area 60 square inches. a) Find its radius. b) Find its circumference. 9) Find the area of the square in the figure below. The circumference of the circle is 18π inches. 10) Find the area of the square in the figure below. The area of the circle is 6π square inches.

19 Special Triangles Three special triangles consistently arise in math classes up to and including calculus. The properties of these triangles are important to learn and memorize so that later work can be done with speed. The triangles are equilateral, isosceles right, and triangles. The equilateral triangle has three congruent sides and therefore has three congruent angles of An angle bisector of an angle cuts the triangle in half and is also perpendicular to a side. s s s s s So, if a side of the equilateral triangle is s, the base is cut in half with each part being s long. The altitude must be s long, since a right triangle is formed with a leg of s and hypotenuse of s. An additional property for the equilateral triangle is as follows. If the side of an equilateral triangle is s the altitude is s. Also, the angles the triangle halves are 0 0, 60 0, and So, the triangle can be summarized below Triangle 0 0 s s Example 1: If the hypotenuse of a triangle is 0, find the other sides. s 60 0 =. Solution: The shortest leg is 0 10 = and the longest leg is 0 10 Example : If the longest leg of a triangle is 15, find the other leg and the hypotenuse. Solution: The longest leg is hypotenuse is s or 0. s long. So, s = 15 and solving s=15. So, the shortest leg is 15 and the Example : If the longest leg is 8 15, find the shortest leg and the hypotenuse.

20 Special Triangles Solution: Since using the triangle above s = 8 15, s can be found by dividing both sides of the equation by. So, s 8 15 = or s = 8 5 and the shortest side is 8 5. The longest side is then (8 5) 16 5 Isosceles Right Triangle =. An isosceles right triangle has two congruent legs. And since the triangle is isosceles, the acute angles are congruent. Therefore, the angles of an isosceles right triangle are If one of the legs is s long and the hypotenuse is h long, then s +s =h. Solving for h. s = h s = h s = h The above properties of the isosceles right triangle are summarized in the figure below s s 45 0 s Example 1: One of the legs of an isosceles triangle is 15. Find the other leg and hypotenuse. Solution: The second leg is the same length 15. The hypotenuse is 15. Example : The hypotenuse of an isosceles triangle is 0. Find the length of the legs. Solution: Using the triangle above, if s = 0, then solving for s will give then leg length. So, the legs are both 15 s = 0 s 0 = s = = = s = 15

21 Classwork Special Triangles Name Given the following side of a triangle, sketch the triangle and find the other sides. Then find the area and perimeter of each triangle. Leave your answers in radical form, simplified as much as possible. 1. Shortest leg: 40. Shortest leg: 6 Area Area Perimeter Perimeter. Hypotenuse: Hypotenuse: 16 Area Area Perimeter Perimeter 5. Longest leg: 5 6. Longest leg: 1 Area Area Perimeter Perimeter Directions: Given the following side in an isosceles right triangle, sketch the triangle and find the remaining sides. Then find the area and perimeter of each triangle. 7. Leg: 7 8. Leg: 5 Area Area Perimeter Perimeter 9. Hypotenuse:5 10. Hypotenuse: 4 6 Area Area Perimeter Perimeter

22 Find x, y, and z in the triangles below z z x x y 60 0 y If a square has a side of 1 inches, find the length of the diagonal. 14. If the diagonal of a square is 0 cm, find the length of a side to the nearest tenth cm. 15. If an equilateral triangle has an altitude of 8, find its area. 16. Find the altitude of the isosceles triangle below. 1cm cm

23 LS 10A Review for Exam 4 Name Simplify each radical as much as possible: ( 5 7)( 5 + 9). 10 ( ) ( 9 5)( 9 + 5) 10. y 198y 7

24 Solve each equation. 11. z = t = 8t + 1. (5x ) = t + 6 = 1t = 1. ( 7x + ) w = x 11x + 10 = y 1 = y 6y = 0. a = 5a 4

25 x = 4 For the following applications, write all radical answers in simplified form and as a decimal rounded to two places.. y = 8y 5. The diagonal of a rectangle is 18 feet. The length is seven times its width. Draw the rectangle and label its diagonal and sides. Use the Pythagorean Theorem to write an equation to find the width of the rectangle. Use your answer to find the length of the rectangle, its area and its perimeter.. w + = w 4. x 7 = The length of a rectangle is four times its width. The area of the rectangle is 180 square meters. Draw the rectangle and label its length and width. Find the length and width of the rectangle. Write an equation for the area of the rectangle and use it to solve the problem. Then find the perimeter of the rectangle and the length of its diagonal.

26 A rectangle has length two more than three times its width. The area of the rectangle is 90 square meters. Find the length and width of the rectangle. Draw the rectangle and label its length and width. Write an equation for the area of the rectangle. Solve your equation to find the solution to this problem. Then find the perimeter of the triangle and the length of its diagonal. 9. One leg of a right triangle is 7 inches longer than the other leg. The hypotenuse of the triangle is 8 inches. Find the length of the legs of the triangle. Draw the triangle and label the sides. Use the Pythagorean Theorem to write an equation. Solve the equation to find the length of the legs. Then use your answers to find the perimeter and area of the triangle. 8. Solve the equation: ( x 1) + ( x + ) = The perimeter of a square is 7 meters. Find the length of each side of the square. Write an equation for the perimeter of the square. Solve the equation to find the length of a side of the square. Then find the area of the square and the diagonal of the square.

27 The diagonal of a square is 18 meters long. Draw the square and label the diagonal. Find the length of a side. Use the Pythagorean Theorem to write an equation. Solve the equation to find the length of a side of the square. Then find the area and the perimeter of the square. 4. A circle has circumference 8 meters. Find its radius. Find its area. 5. A circle has area 49π square meters. Find its radius. Find its circumference.. A circle has diameter 0 inches. Find its circumference. Find its area.. A circle has circumference 8 π inches. Find its radius. Find its area. 6. A circle has area 49 square meters. Find its radius. Find its circumference.

28 The shortest leg of a triangle is 5 inches. Draw the triangle and label the angles and sides. Find the length of the other leg and the hypotenuse. Find the perimeter and area of the triangle The longest leg of a triangle is 17 feet. Draw the triangle and label the angles and sides. Find the length of the other leg and the hypotenuse. Find the perimeter and area of the triangle The hypotenuse of a triangle is14 feet. Draw the triangle and label the angles and sides. Find the length of the two legs. Find the perimeter and area of the triangle. 40. Together Ally and Lauren can paint the living room in twelve hours. Working alone Ally can paint the living room in four hours more than it takes Lauren working alone. How long will it take each of them, working alone, to paint the living room? Write an equation and solve it. Round your answers to the nearest two decimal places.

29 Answers to Review for Exam ± 5 1 ± or or 8. 1 or ± 18 = ± y 4 y 11. ± or ± or ± 96 ± = ± = ± or x + x = ( 7x) 9 5 = 18 ft.55 ft 6 l = ft 17.8 ft P = ft 40.7 ft 5 A = 45.6 square feet 4w w = P = 0 d = = 180 l = ± w = P = 45. m d = m 5 m 6.71m w(w + ) = 90 5 m m m = 7.66 m 1± 71 = 5.15 m

30 ± 456 ± 114 x = = or r = 7 m C = 14π 4.96 m 9. x + ( x + 7) 14 ± 11,56 7 ± x = = 4 other leg is 0.14 in P = 91.8 in = 8 A = 48.7 square inches 0. s = 18 m A = 4 m d = 18 m s = 9 P = 6 m A = 16 m m C = 0π 94. inches A = 5π in r = 14 in A = 196π in r 4.46 m A 6.46 m in r.95 m C 4.81 m long leg hypotenuse Perimeter Area long leg Area 5 short leg Perimeter 49 short leg hypotenuse Perimeter Area = x x in 10 in ( in 7 7 ft ( ft 4 ft ft ( Ally 6.17 hours ft ft ) in ) ) x = = Lauren.17 hours ft ft 7.17 hrs