Teacher resources and professional development across the curriculum

Teacher professional development and classroom resources across the curriculum

Solutions for Session 5, Part C

See solutions for Problems: C1 | C2 | C3 | C4 | C5 | C6 | C7 | C8 | C9
C10 | C11 | C12 | C13

Problem C1

 a. If the ladder is too steep, it may be difficult to climb, and there is a good chance the ladder will fall over backward (say, in a strong wind). b. If the ladder is not steep enough, it may also be difficult to climb, it is likely to fall forward, and it may not reach high enough to be useful.

 Problem C2 Answers will vary. You should find that as the angle between the ground and the ladder increases, the height that the ladder reaches on the wall increases while the distance from the base decreases.

 Problem C3 When the angle is 45 degrees, the height and distance are equal. When the angle is larger than 45 degrees, the height-to-distance ratio is greater than 1, and when the angle is smaller than 45 degrees, the ratio is less than 1. Importantly, this ratio is based entirely on the angle, rather than on the length of the actual ladder used.

 Problem C4 As the ratio increases, the angle increases, but it will always be less than 90 degrees.

 Problem C5 By common definition, height is measured along a line that is perpendicular to the base, so the angle must be 90 degrees. If the angle were not 90 degrees, we would not be measuring the vertical height of the ladder against the wall, but some other distance -- which would also affect the height-to-distance ratio.

Problem C7

Answers may vary. Here are some possibilities:

Problem

Measure

h:d Ratio

Ratio as Decimal

 a 45° 5:5 (or 1:1) 1 b 63° 6:3 (or 2:1) 2 c 30° 3.5:6 0.58 d 27° 3:6 (or 1:2) 0.5 e 60° 6:3.5 1.72

 Problem C8 The ratios are reciprocals (2:1 and 1:2), while the angles are complementary (they sum to 90 degrees). One way to think about this is that if we "reversed" the triangle (switched h and d), we should also reverse the angles in the triangle. The 90-degree angle remains fixed, so the other two angles will switch. Since they are complementary, if one is 63 degrees, the other is 27 degrees, and vice versa. While it might be easy to see that the measures of both angles sum to 90 degrees, seeing that the h:d ratios are inverses may not be as obvious. Notice that the angles in Problem C7 (c) and (e) are 30 and 60 degrees respectively, and are also complementary angles.

Problem C9

 a. Here is the completed Steepness Graph: b. The h:d ratio increases as the measure of increases, but the ratio is increasing at a greater rate. As the angle approaches 90 degrees, the ratio grows increasingly large (with no limit!). Try drawing a triangle with an 85-degree angle and then measure the h:d ratio. It will be very large! Notice how this is shown on the graph where the curve becomes steeper after the 45-degree mark.

 Problem C10 This range can be determined by drawing triangles or by referring to a table of values for these ratios. The smallest safe angle is about 63 degrees (see Problem C7, part b), while the largest is about 72 degrees.

 Problem C11 Tan 52° = b/30, or 1.28= b/30. Multiplying both sides by 30 yields the width of the river, which is 38.4 m.

 Problem C12 One approach is to rewrite these ratios as unit ratios and then compare them. The ratios are as follows: Glider 1 -- 1:27 Glider 2 -- 1:25 Glider 3 -- 1:26 Glider 1 has the smallest glide ratio, so it can travel farther (27 m for every 1 m that it descends), and it descends at the slowest rate; therefore, it is the safest. Another approach is to convert the ratios to decimals and then compare them (this time, looking for the smallest decimal).

 Problem C13 Using a calculator, we see that tan 35° = 0.70. Since we know that tan 35° = h/d, we just plug in the numbers: 0.70 = 100/d The distance is 100/0.7, or approximately 143 m.