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Measurement Session 5: Solutions
 
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A B C 
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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.

<< back to Problem C1


 

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.

<< back to Problem C2


 

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.

<< back to Problem C3


 

Problem C4

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

<< back to Problem C4


 

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.

<< back to Problem C5


 

Problem C6

Answers will vary. (Tan 30° is about 0.58, while tan 60° is about 1.73.)

<< back to Problem C6


 

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

<< back to Problem C7


 

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.

<< back to Problem C8


 

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.

<< back to Problem C9


 

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.

<< back to Problem C10


 

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.

<< back to Problem C11


 

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).

<< back to Problem C12


 

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.

<< back to Problem C13

 

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