Math / Trigonometry

QuestionExample 6: Emma is on a 50 meter high bridge and sees two boats anchored below. From her position, boat A has a bearing of $230^{\circ}$ and boat $B$ has a bearing of $120^{\circ}$ . Emma estimates the angles of depression to be $38^{\circ}$ for boat A and $35^{\circ}$ for boat B. How far apart are the boats to the nearest meter?

Studdy Solution

STEP 1

1. The height of the bridge is $50$ meters.
2. The angles of depression to boats A and B are $38^{\circ}$ and $35^{\circ}$ respectively.
3. Bearings are measured clockwise from the north direction.

STEP 2

1. Understand the relationship between angles of depression and horizontal distances.
2. Calculate the horizontal distances from Emma to each boat.
3. Use trigonometry to find the positions of the boats relative to each other.
4. Calculate the distance between the two boats using the Law of Cosines.

STEP 3

Understand that the angle of depression from Emma to a boat is equal to the angle of elevation from the boat to Emma. This means we can use trigonometry to find the horizontal distance from Emma to each boat.

STEP 4

Calculate the horizontal distance from Emma to boat A using the angle of depression:
Let $d_A$ be the horizontal distance to boat A. $\tan(38^{\circ}) = \frac{50}{d_A}$ $d_A = \frac{50}{\tan(38^{\circ})}$
Calculate $d_A$ .

STEP 5

Calculate the horizontal distance from Emma to boat B using the angle of depression:
Let $d_B$ be the horizontal distance to boat B. $\tan(35^{\circ}) = \frac{50}{d_B}$ $d_B = \frac{50}{\tan(35^{\circ})}$
Calculate $d_B$ .

STEP 6

Use trigonometry to find the positions of the boats relative to each other. The bearings provide angles with respect to the north direction:
- Boat A is at a bearing of $230^{\circ}$ . - Boat B is at a bearing of $120^{\circ}$ .
Calculate the angle between the two bearings:
$\text{Angle between bearings} = 230^{\circ} - 120^{\circ} = 110^{\circ}$

STEP 7

Use the Law of Cosines to find the distance between the two boats:
Let $c$ be the distance between the boats. $c^2 = d_A^2 + d_B^2 - 2 \cdot d_A \cdot d_B \cdot \cos(110^{\circ})$
Calculate $c$ and round to the nearest meter.
The distance between the boats is:
$\boxed{c \text{ meters}}$

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Studdy Solution

STEP 1

1. The height of the bridge is $50$ meters.
2. The angles of depression to boats A and B are $38^{\circ}$ and $35^{\circ}$ respectively.
3. Bearings are measured clockwise from the north direction.

STEP 2

1. Understand the relationship between angles of depression and horizontal distances.
2. Calculate the horizontal distances from Emma to each boat.
3. Use trigonometry to find the positions of the boats relative to each other.
4. Calculate the distance between the two boats using the Law of Cosines.

STEP 3

Understand that the angle of depression from Emma to a boat is equal to the angle of elevation from the boat to Emma. This means we can use trigonometry to find the horizontal distance from Emma to each boat.

STEP 4

Calculate the horizontal distance from Emma to boat A using the angle of depression:
Let $d_A$ be the horizontal distance to boat A. $\tan(38^{\circ}) = \frac{50}{d_A}$ $d_A = \frac{50}{\tan(38^{\circ})}$
Calculate $d_A$ .

STEP 5

Calculate the horizontal distance from Emma to boat B using the angle of depression:
Let $d_B$ be the horizontal distance to boat B. $\tan(35^{\circ}) = \frac{50}{d_B}$ $d_B = \frac{50}{\tan(35^{\circ})}$
Calculate $d_B$ .

STEP 6

STEP 7

Use the Law of Cosines to find the distance between the two boats:
Let $c$ be the distance between the boats. $c^2 = d_A^2 + d_B^2 - 2 \cdot d_A \cdot d_B \cdot \cos(110^{\circ})$
Calculate $c$ and round to the nearest meter.
The distance between the boats is:
$\boxed{c \text{ meters}}$

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Studdy Solution

STEP 1

1. The height of the bridge is 50 50 50 meters.2. The angles of depression to boats A and B are 38∘ 38^{\circ} 38∘ and 35∘ 35^{\circ} 35∘ respectively.3. Bearings are measured clockwise from the north direction.

STEP 2

1. Understand the relationship between angles of depression and horizontal distances.2. Calculate the horizontal distances from Emma to each boat.3. Use trigonometry to find the positions of the boats relative to each other.4. Calculate the distance between the two boats using the Law of Cosines.

STEP 3

Understand that the angle of depression from Emma to a boat is equal to the angle of elevation from the boat to Emma. This means we can use trigonometry to find the horizontal distance from Emma to each boat.

STEP 4

STEP 5

STEP 6

STEP 7

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Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. The height of the bridge is $50$ meters.
2. The angles of depression to boats A and B are $38^{\circ}$ and $35^{\circ}$ respectively.
3. Bearings are measured clockwise from the north direction.