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Ratio and Proportion Lab An investigation using Shadows and Mirrors Name ______________ Directions: Complete each problem as indicated at its station. For each problem, 1. Draw a picture, 2. Make a similarity statement, telling which triangles are similar to each other, 3. Label all parts with correct measurements, 4. Write an equation showing the proper proportions, 5. Solve the problem. 6. Make all measurements carefully to the nearest mm (If doing a word problem, use feet and inches), and 7. Write all answers in complete sentences. D Use the following variables at the Mirror Measurement stations: d = distance from the floor to eye level f = distance from your feet to the reflection of the object in the A mirror h w = distance from the reflection in the mirror to the wall h = the height from the floor to the object d *Make all measurements to the nearest mm, and write all answers in complete sentences. C f B w E Station 1: Marcus is looking down into a mirror (see illustration) and has moved back to see the top of the lamp post at the top edge of the mirror. He knows that his eyes are 6’ off of the ground and he is 1.5’ from that point on the mirror. The distance from that point on the mirror to the lamp post is 37 ½” . How tall is the lamp post? (use inches/feet) Station 2: Measure the height of the __________________________(use cm) Drew Moore and Nancy Powell, Bloomington High School, Bloomington, IL - NCTM, 2006 Indirect Measurement - Page 1 of 7 Station 3: Jared is standing 5’ in front of a tree and he is 6’ tall. His shadow is 4 foot long. How tall is the apple tree? (use inches/feet) Station 4: Measure the height of the __________________________(use cm) Station 5 One day the shadow of the small 3’ 4” tree cast a shadow of 5 feet. The palm tree cast a shadow of 25.5 feet. How tall is the palm tree? (use inches/feet) Station 6: Measure the height of the __________________________(use cm) Drew Moore and Nancy Powell, Bloomington High School, Bloomington, IL - NCTM, 2006 Indirect Measurement - Page 2 of 7 Station 7: The light is sitting on the floor behind the chair at a distance of 1 ½ feet from the bottom of the lamp to the back of the chair. The lamp is shining on the chair and casts a shadow 4 foot in front of the chair and the light source is 66 inches off the floor over the bottom of the lamp. How tall is the back of the chair? (use inches/feet) Station 8: Measure the height of the __________________________(use cm) Station 9 If the sign is 5’ l0” tall and is 1’ 5” from the lamp post which is 8’ 8” tall, how long is the sign’s shadow? (use inches/feet) Drew Moore and Nancy Powell, Bloomington High School, Bloomington, IL - NCTM, 2006 Indirect Measurement - Page 3 of 7 Station 10: Johnny is looking through a magnifying glass (the magnifying glass is parallel to the ground) at a bug. He measured the bug’s image on the magnifying glass and you know that the magnifying glass is 3 cm from the object. On the magnifying glass, the bug measures 4cm long by 2 cm wide. Before it flew away he measured the bug to be .75 cm long. Help Johnny by calculating the width of the bug. Drew Moore and Nancy Powell, Bloomington High School, Bloomington, IL - NCTM, 2006 Indirect Measurement - Page 4 of 7 NAME: ____________________________ Angle of Elevation Indirect measurement with a Clinometer Each group will need: - tape measure - clinometer - sidewalk chalk It is not always possible to measure the height of a tall object by running a tape measure along its length and so we must measure it indirectly by measuring other quantities directly and using the proper mathematics to calculate the height. We are going to use a clinometer to measure angles of elevation from our line of sight to the top of a tall object. Suppose we wanted to measure how tall a building is, consider the following diagram: θ Eye level observe r ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ Ground level 1. The angle of elevation, θ, is the angle that we can measure with the clinometer. What other quantities can we measure directly? Label these on the diagram. Take all measurements in inches. 2. Considering your answers to question #1, which trig ratio will allow you to measure the height of the building? 3. Why is it important to measure from the ground to your eyes? 4. Using four different locations, two per partner, you must find the height of the school. Draw pictures making sure to label the measurements that you’ve taken and show all of your calculations. What did you find? Is the height of the building the same looking at it from each location? 5. Find the height of the main building (the part south of our grid where the glass doors are). Drew Moore and Nancy Powell, Bloomington High School, Bloomington, IL - NCTM, 2006 Indirect Measurement - Page 5 of 7 NAME : _____________________________ Angle of Depression Mission: Measure the height of the first flight of stairs in feet and inches outside Mrs. Powell’s room. You are here A D E h B C LABEL YOUR PICTURE !! ** Make sure you spot where the wall meets the floor for best accuracy. Your clinometers will give you the angle of depression. ** 1. How far away are you from the wall? Or how far are you from the bottom of the wall? Where should you measure from? 2. What is the angle of depression to the bottom of the wall? What is its measure? 3. Now consider carefully the angle of depression and the angle you want to use. Are they the same angle? If not, are there congruent angles in the picture? 4. Since you want to find the height of the stairs, and have the information from question 1, 2, and question 3, which trig ratio should I use? 5. Now you’re ready to find the height. Be careful! 6. Is the height of the stairs the same as the side of the triangle you are using? Show all work. Drew Moore and Nancy Powell, Bloomington High School, Bloomington, IL - NCTM, 2006 Indirect Measurement - Page 6 of 7 Name ______________________________________ Hour_______ Directions: Use what you know about similar figures and the ratio of their sides to find the answers to these questions. You’re going to build a Boeing 747-400 model airplane using a scale of 1:144. 1. What does it mean that the model’s scale is 1:144? 2. Using the dimensions below, find the measures in feet and inches, and round the inches to the nearest eighth of an inch. Show work to justify your answer and label each measurement with correct units. a. Length of the model b. Wing span of the model c. Height of the model d. Tail span of the model 3. Using the metric dimensions below, find the measures to the nearest millimeter (tenth of a Centimeter). Show work to justify your answer and label each measurement with correct units. a. Length of the model b. Wing span of the model c. Height of the model d. Tail span of the model 4. If another model of the same plane has a length of 4’10” , what is the scale of this other model? (give your answer in fraction form: 1: __?__ ) 5. If you were given a choice whether to use English units (feet and inches) or Metric units (meters and centimeters), which would you choose and why? Drew Moore and Nancy Powell, Bloomington High School, Bloomington, IL - NCTM, 2006 Indirect Measurement - Page 7 of 7