d Connect the plotted points with a dashed line to find the prediction.
A
a See solution.
B
b
Angle of depression
Length of line of sight (feet)
40^(∘)
≈ 46.7
50^(∘)
≈ 39.2
60^(∘)
≈ 34.6
70^(∘)
≈ 31.9
80^(∘)
≈ 30.5
C
c
D
d 60 feet
Practice makes perfect
a We are given that we are standing on a 30-foot cliff and we see a sailboat. Let's draw a diagram of this situation. We will assume that the cliff is perpendicular to the ocean.
b In this part we want to make a table showing the angle of depression. We will call it α and the length of our line of sight l. Let's recall that the angle of depression is the angle between the line of sight and the horizontal distance to the object.
Notice that we can write an equation using the definition of the sine of an angle. Recall that in a right triangle the sine of an angle is the ratio between the leg opposite to this angle and the hypotenuse.
sin α=30/l ⇓ l=30/sin α
Now we will substitute the given angle measures into the above formula and solve for l using the calculator. We will record the results in the table. All results will be rounded to one decimal.
α
30/sin α
l
40^(∘)
30/sin 40^(∘)
≈ 46.7
50^(∘)
30/sin 50^(∘)
≈ 39.2
60^(∘)
30/sin 60^(∘)
≈ 34.6
70^(∘)
30/sin 70^(∘)
≈ 31.9
80^(∘)
30/sin 80^(∘)
≈ 30.5
c Next we will graph the values we found in the previous part. Let the angle of depression α be on the horizontal axis and the approximate length of the line of sight on the vertical axis.
d Finally we are asked to predict the length of the line of sight when the angle of depression is 30^(∘). To do this we will connect the graphed points with a curve and extend it to meet the line α=30^(∘).
We can predict that the length of the line of sight when the angle of depression is 30^(∘) will be approximately 60 feet.
