A ship is headed directly toward a coastline formed by a vertical cliff (segment BC) that is 80 meters high. At point A, from the ship, the measure of the angle of elevation to the top of the cliff (B) is 10°. A few minutes later, at point D, the measure of the angle of elevation to the top of the cliff has increased to 20°.

Question

A ship is headed directly toward a coastline formed by a vertical cliff (segment BC) that is 80 meters high. At point A, from the ship, the measure of the angle of elevation to the top of the cliff (B) is 10°. A few minutes later, at point D, the measure of the angle of elevation to the top of the cliff has increased to 20°.

Determine the following to the nearest meter.

Segment DC
Segment AC

Answer 1

DC=220 meters

AC=454 meters

Answer 2

tan 20 = 80/DC

tan 10 = 80/AC

DC = 80 / tan 20
AC = 80 / tan 10

Answer 3

To solve this problem, we can use trigonometric ratios. Let's start by labeling the given angles and lengths on the diagram.

C
|\
80 | \
| \
| \
| \ A
| \
|10° \20°
|______\
D B

Segment AC is the distance from the ship at point A to the base of the cliff at point C.

Segment DC is the distance from the ship at point D to the base of the cliff at point C.

We can use the tangent function to find the lengths of these segments.

1. Segment AC:
In triangle ABC, we can use the tangent of angle 10° to find the length of segment AC.
tan(10°) = opposite/adjacent
tan(10°) = 80/AC

Rearranging the equation to solve for AC:
AC = 80 / tan(10°)
AC ≈ 459 meters (rounded to the nearest meter)

2. Segment DC:
In triangle DBC, we can use the tangent of angle 20° to find the length of segment DC.
tan(20°) = opposite/adjacent
tan(20°) = 80/DC

Rearranging the equation to solve for DC:
DC = 80 / tan(20°)
DC ≈ 231 meters (rounded to the nearest meter)

Therefore, the lengths of segment AC and DC are approximately 459 meters and 231 meters, respectively.

Answer 4

To determine the lengths of segments DC and AC, we can use trigonometric ratios.

In triangle ABC, segment DC represents the increase in distance between the ship and the cliff, and segment AC represents the total distance between the ship and the cliff.

First, let's calculate the length of segment DC.

In triangle ABC, the angle of elevation at point A is 10°, and the vertical height of the cliff (segment BC) is 80 meters. We can use the tangent function to find the length of segment DC.

The tangent of an angle is defined as the ratio of the length of the opposite side to the length of the adjacent side. In this case, the opposite side is the vertical height of the cliff (80 meters), and the adjacent side is segment DC.

So, we have: tan(10°) = 80/DC

Now, isolate DC: DC = 80 / tan(10°)

Using a scientific calculator, calculate the tangent of 10°: tan(10°) ≈ 0.1763269807

Now, substitute this value back into the equation: DC ≈ 80 / 0.1763269807

Calculate the value: DC ≈ 453.979 m

So, to the nearest meter, segment DC is approximately 454 meters.

Now, let's calculate the length of segment AC.

In triangle ACD, the angle of elevation at point D is 20°, and we already know the length of segment DC (approximately 453.979 meters).

Using the tangent function again, we have: tan(20°) = 80/AC (using the vertical height of the cliff)

Isolate AC: AC = 80 / tan(20°)

Using a scientific calculator, calculate the tangent of 20°: tan(20°) ≈ 0.3644710503

Now, substitute this value back into the equation: AC ≈ 80 / 0.3644710503

Calculate the value: AC ≈ 219.042 m

So, to the nearest meter, segment AC is approximately 219 meters.