Learning Math: Measurement
Indirect Measurement and Trigonometry Homework
Session 5, Homework
At a distance of 160 m from a tower, you look up at an angle of 23 degrees and see the top of the tower:
What is the height of the tower?
Compute the height of this extremely steep road at point C for the drawing below:
Draw a side view of the flight path for a glider whose glide angle is 5 degrees. What is the glide ratio?
One glider has a glide ratio of 1:40, while another has a glide angle of 3 degrees. Which glider flies farther? Explain why.
Suppose that a glider has a glide ratio of 1:40. It is flying over a village at an altitude of 230 m, and it’s 9 km from an airstrip. Can it reach the airstrip? Explain.
An electricity line pole makes an angle of 75 degrees with the road surface, as shown below:
How much does the road rise over a horizontal distance of 100 m?
- Your friend places a mirror 30 ft. from the base of a tall tree. Then she steps back from the mirror until she sees the top of the tree in the mirror’s center. What can be said about the angle formed from the treetop to the mirror to the base of the tree, and the angle formed from her head to the mirror to the base of her feet? What do you know about the other angles in the triangles formed below?
- How might this information be used to determine the height of the tree?
- You know that your friend is 6 ft. tall and that the mirror is 30 ft. from the base of the tree. After your friend moves back 4 ft. from the mirror, she can see the treetop’s reflection. How tall is the tree?
Take It Further
Pretend that you are standing at the equator at noon one day, and the Sun’s rays are directly overhead (casting no shadow). Meanwhile, your friend, who is located 787 km away, calls you and tells you that at that very moment the Sun is casting a shadow, and that he had measured and calculated that the Sun’s rays are coming in at 7.2 degrees. Knowing that there are 360 degrees around the Earth from its center point, use this information to estimate the Earth’s circumference. Compare this estimate of the Earth’s circumference to today’s known value of 40,075.16 km.
Draw some pictures. The 7.2-degree angle will be opposite the length of 787 km. Fifty of these 7.2-degree angles give a complete circle.
This problem has a great deal of history behind it. Eratosthenes is credited with using this approach to calculate the circumference of the Earth. He concluded that the only explanation for why no shadows fell at Syene at midday on June 21 but did fall at Alexandria was because of the curvature of the Earth. He then devised a method for finding the circumference of the Earth that is based on constant-ratio calculations involving proportions. On the day of the summer solstice, he measured the direction of the Sun’s rays as they struck an obelisk in Alexandria and an angle between them. This angle (we’ll call it ∠A) can be compared to 360 degrees, and this ratio can in turn be used in the following proportion:
∠A = Distance Between Alexandria and Syene
360° Circumference of Earth
He measured ∠A to be 8 degrees and the distance from Alexandria to Syene as 4,800 Greek stadia (a stadia corresponds to approximately 606.75 ft.). He then set up a proportion similar to the one above. Eratosthenes’s approximate measurement for the circumference of the Earth was very close to today’s modern value.
Using a calculator, we see that tan 23° = 0.42. Therefore, 0.42 = h/160, so h = 160 • 0.42, or 67.2 m.
Using a calculator, we see that tan 12° = 0.21. Therefore, 0.21 = h/10, so h = 10 • 0.21, or 2.1 m.
The tangent of a 5-degree angle is 0.0875. This is a glide ratio of about 1:11.4, so the glider flies 11.4 m for every meter it drops.
The tangent of a 3-degree angle is 0.0524. This is a glide ratio of about 1:19, which is much less than 1:40, so the glider with ratio 1:40 flies more than twice as far.
The distance the glider can travel is 230 • 40 = 9,200 m, or 9.2 km. So yes, the glider can reach the airstrip.
The angle of 75 degrees means that the road rises at a 15-degree angle. Tan 15o is about 0.27, which equals h/100; therefore, h is about 27 m.
- The two angles are equal. They are known as the angle of incidence and the angle of reflection, and from physics we know they are equal. Also, since we know they are both right triangles, the angle at the top of the tree is equal to the angle at your friend’s head.
- Since we’ve determined that the angles in the triangles are the same, they are similar triangles. We can set up the following proportion to help us find the height of the tree:
- Using this proportion, we get the following:
x/6 = 30/4
The height of the tree is 45 ft.
Since 50 of these 7.2-degree angles give a complete circle and we know the length between where the Sun is overhead and where it is at an angle, we can use this to approximate the circumference of the Earth:
360 ÷ 7.2 = 50
50 • 787 = 39350 km
It’s pretty close!
Session 1 What Does It Mean To Measure?
Explore what can be measured and what it means to measure. Identify measurable properties such as weight, surface area, and volume, and discuss which metric units are more appropriate for measuring these properties. Refine your use of precision instruments, and learn about alternate methods such as displacement. Explore approximation techniques, and reason about how to make better approximations.
Session 2 Fundamentals of Measurement
Investigate the difference between a count and a measure, and examine essential ideas such as unit iteration, partitioning, and the compensatory principle. Learn about the many uses of ratio in measurement and how scale models help us understand relative sizes. Investigate the constant of proportionality in isosceles right triangles, and learn about precision and accuracy in measurement.
Session 3 The Metric System
Learn about the relationships between units in the metric system and how to represent quantities using different units. Estimate and measure quantities of length, mass, and capacity, and solve measurement problems.
Session 4 Angle Measurement
Review appropriate notation for angle measurement, and describe angles in terms of the amount of turn. Use reasoning to determine the measures of angles in polygons based on the idea that there are 360 degrees in a complete turn. Learn about the relationships among angles within shapes, and generalize a formula for finding the sum of the angles in any n-gon. Use activities based on GeoLogo to explore the differences among interior, exterior, and central angles.
Session 5 Indirect Measurement and Trigonometry
Learn how to use the concept of similarity to measure distance indirectly, using methods involving similar triangles, shadows, and transits. Apply basic right-angle trigonometry to learn about the relationships among steepness, angle of elevation, and height-to-distance ratio. Use trigonometric ratios to solve problems involving right triangles.
Session 6 Area
Learn that area is a measure of how much surface is covered. Explore the relationship between the size of the unit used and the resulting measurement. Find the area of irregular shapes by counting squares or subdividing the figure into sections. Learn how to approximate the area more accurately by using smaller and smaller units. Relate this counting approach to the standard area formulas for triangles, trapezoids, and parallelograms.
Session 7 Circles and Pi (π)
Investigate the circumference and area of a circle. Examine what underlies the formulas for these measures, and learn how the features of the irrational number pi (π) affect both of these measures.
Session 8 Volume
Explore several methods for finding the volume of objects, using both standard cubic units and non-standard measures. Explore how volume formulas for solid objects such as spheres, cylinders, and cones are derived and related.
Session 9 Measurement Relationships
Examine the relationships between area and perimeter when one measure is fixed. Determine which shapes maximize area while minimizing perimeter, and vice versa. Explore the proportional relationship between surface area and volume. Construct open-box containers, and use graphs to approximate the dimensions of the resulting rectangular prism that holds the maximum volume.
Session 10 Classroom Case Studies, K-2
Watch this program in the 10th session for K-2 teachers. Explore how the concepts developed in this course can be applied through case studies of K-2 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for K-2 students.
Session 11 Classroom Case Studies, 3-5
Watch this program in the 10th session for grade 3-5 teachers. Explore how the concepts developed in this course can be applied through case studies of grade 3-5 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for grade 3-5 students.
Session 12 Classroom Case Studies, 6-8
Watch this program in the 10th session for grade 6-8 teachers. Explore how the concepts developed in this course can be applied through case studies of grade 6-8 teachers (former course participants who have adapted their new knowledge to their classrooms), as well as a set of typical measurement problems for grade 6-8 students.