the questions below. Make sure to show your work and provide complete geometric explanations for full credit. 1. A triangle has side lengths 10, 15, and 7. Is the triangle acute, obtuse, or right? Show your work and explain. 7, 10, 15 [arrange smallest to largest] a2 +b 2=c 2 [set up the equation using the Pythagorean theorem] 2 2 7 +10 =15 2 [substitute the values into the equation] 49+100=225 [evaluate the power] 149=225 [add the numbers] To know which triangle it is, check c and see if it is greater or less than a and b. The triangle is obtuse. 2. A highway makes an angle of 6 with the horizontal. This angle is maintained for a horizontal distance of 5 miles. To the nearest hundredth of a mile, how high does the highway rise in this 5-mile section? Show the steps you use to find the distance. x tan 6 = 5 x=5tan 6 x 0.53 miles 3. A forest ranger spots a fire from a 28-foot tower. The angle of depression from the tower to the fire is 11 . To the nearest foot, how far is the fire from the base of the tower? Show the steps you use to find the solution. tan 11 = x= 28 x 28 tan 11 [tangent ratio] [solve for x] x 144 The fire is about 144 feet from the vase of the tower. 4. State whether the transformation appears to be a rigid motion. Explain. Rigid transformation that do not change the shape or size of the pre-image and non-rigid transformation that change the size but not the shape of the pre-image. Here compared to pre-image there is change in size and no change in shape. So, the pre-image got the non-rigid transformation. This is a non-rigid transformation. It's a non-rigid motion. 5. How can you verify Euler's formula for this net of a cube? Closed this net would be a cube. In a cube there are 8 vertices, 6 faces and 12 edges. F+V-E is 8+6-12=2 thus this net cube holds true for Euler's formula. 6. Jason designed an arch made of wrought iron for the top of a mall entrance. The 11 segments between the two concentric circles are each 1.25 m long. Find the total length of wrought iron used to make the structure. Round the answer to the nearest meter. Show your work. There are 11 segments between two concentric circles. Each segment is 1.25m long. Total wrought iron used for 11 segments are obtained by multiplying 11 by 1.25. This equals 13.75m. The two concentric circles are also made of wrought iron. These are the shape of a semi-circle without the straight edge. Given the arch extends up to 15m, subtract 1.25m twice for the segments. Thus the inner circle has diameter of 12.5 and the outer circle has diameter of 15m. Thus the total length of wrought iron used is the sum of the eleven segments, plus the sum of the inner circumference, plus the sum of the outer circumference. 1.25 1 1 C + C +11 ) 2 o 2 i 1 1 15+ 12.5+11 ( 1.25 ) 2 2 7.5 +6.25 +13.75 13.75 +13.75 57 The length of wrought iron used is approximately 57 meters. 7. Consider the prism shown below. Find the surface area of the prism. Show your work T.S.A.=ph+2b [p represents the perimeter of the base, h the height of the prism and b the area of the base] Perimeter of base = 10+6+8=24m Height of prism = 17m Area = 1 68=24 m 2 Surface area = 24*17+24 =24*18 = 432 m 2 8. Janine made a cylindrical vase in which the sum of the lateral area and area of one base was about 3000 square centimeters. The vase had a height of 50 centimeters. Find the radius of the vase. Explain the method you would use to find the radius. Lateral area of Cylinder = 2 2 rh+ r =3000 cm 2 2 rh and if h=50, we have 2 2 r (50)+ r =3000 cm 2 100 r + r 2=3000 cm 2 divide both sides by 3000 100 r +r 2= subtract 3000 from both sides r 2 +100 r 3000 =0 Reject the negative solution Radius 8.78 9. A design on the surface of a balloon is 5 cm wide when the balloon holds 71 cm of air. How much air does the balloon hold when the design is 10 cm wide? Explain the method you use to find the amount of air. 10. In circle O, a. b. and are tangents. (The figure is not drawn to scale.) Prove that . Find m BOD for m AOP = 64. Explain your reasoning. 11. Determine whether a tangent line to circle O is shown in the diagram, for AB = 7.75, OB = 4, and AO = 8.75. Explain your reasoning. (The figure is not drawn to scale.) 12. In circle O, CD = 44, OM = 20, ON = 19, scale.) Show your work. a. b. c. , (The figure is not drawn to Find the radius. If your answer is not an integer, express it in radical form. Find FN. If your answer is not an integer, express it in radical form. Find EF. Express it as a decimal rounded to the nearest tenth. 13. is tangent to the circle at B. m A = 14 and your work. a. Find x. b. Find y. (The figure is not drawn to scale.) Show 14. At a track meet, 50 people ran the 100-meter dash. 2 people finished in 11 seconds, 5 people finished in 12 seconds, 8 people finished in 13 seconds, 10 people finished in 14 seconds, 21 people finished in 15 seconds, 2 people finished in 16 seconds, and 2 people finished in 17 seconds. What is the probability distribution for the finish times? 15. Is there a similarity transformation that maps to ? If so, identify the similarity transformation and write a similarity statement. Explain your answer. A 4 C D 4 3 B F 8 8 6 E 16. Janine wants to paint just the sides of a cylindrical pottery vase that has a height of 45 cm and a diameter of 14 cm. To the nearest whole number, find the number of square centimeters she will need to paint. Explain the method you would use to find the lateral area. 17. Determine whether a tangent line is shown in the diagram, for AB = 7, OB = 3.75, and AO = 8. Explain your reasoning. (The figure is not drawn to scale.) 18. The equation (x + 5) + (y + 3) = 169 models the position and range of the source of a radio signal. Describe the position of the source and the range of the signals. 19. An hourglass, composed of two identical cones, is 12 cm tall. The radius of each cone is 3 cm. If you want to fill the bottom half of the hourglass full of salt, how much salt will you need? Explain the method you would use to find the amount of salt. 20. Three balls are packaged in a cylindrical container as shown below. The balls just touch the top, bottom, and sides of the cylinder. The diameter of each ball is 13 cm. a. What is the volume of the cylinder? Explain your method for finding the volume. b. What is the total volume of the three balls? Explain your method for finding the total volume. c. What percent of the volume of the container is occupied by the three balls? Explain how you would find the percent