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Draw a right-angled triangle. The cliff is the vertical side, the distance to the boat is the horizontal side, and the line of sight to the boat is the hypotenuse. The angle of depression from the top

Psychology

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2 parts
8

369.08 m\text{369.08 m}

9

86.43 m\text{86.43 m}

Step-by-step solution

9 steps

Part 8· 4 steps

  1. Draw a right-angled triangle.

    The cliff is the vertical side, the distance to the boat is the horizontal side, and the line of sight to the boat is the hypotenuse. The angle of depression from the top of the cliff to the boat is equal to the angle of elevation from the boat to the top of the cliff due to alternate interior angles.

  2. Identify the known values.

    The height of the cliff (opposite side to the angle of elevation) is 180 m180 \text{ m}.
    The angle of elevation is 26∘26^\circ.
    We need to find the horizontal distance from the base of the cliff to the boat (adjacent side).

  3. Use the tangent function, which relates the opposite and adjacent sides to the angle.

    tan⁡(θ)=oppositeadjacent\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} tan⁡(26∘)=180x\tan(26^\circ) = \frac{180}{x}

  4. Solve for xx.

    x=180tan⁡(26∘)x = \frac{180}{\tan(26^\circ)} x≈1800.4877x \approx \frac{180}{0.4877} x≈369.08 mx \approx 369.08 \text{ m}

    The boat is approximately 369.08 m369.08 \text{ m} from the base of the cliff.

    369.08 m\boxed{\text{369.08 m}}

Part 9

86.43 m\text{86.43 m}

  1. Draw a right-angled triangle.

    The horizontal distance from Sarah to the tree is the adjacent side. The vertical height of the tree above Sarah's eye level is the opposite side.

  2. Identify the known values.

    The horizontal distance from Sarah to the base of the tree (adjacent side) is 180 m180 \text{ m}.
    The angle of elevation is 25∘25^\circ.
    Sarah's eye level is 2.5 m2.5 \text{ m} above the ground.

  3. Calculate the height of the tree above Sarah's eye level using the tangent function.

    tan⁡(θ)=oppositeadjacent\tan(\theta) = \frac{\text{opposite}}{\text{adjacent}}

    Let h′h' be the height of the tree above Sarah's eye level.

    tan⁡(25∘)=h′180\tan(25^\circ) = \frac{h'}{180}

  4. Solve for h′h'.

    h′=180×tan⁡(25∘)h' = 180 \times \tan(25^\circ) h′≈180×0.4663h' \approx 180 \times 0.4663 h′≈83.934 mh' \approx 83.934 \text{ m}

  5. Calculate the total height of the tree by adding Sarah's eye level.

    Total height=h′+Sarah’s eye level\text{Total height} = h' + \text{Sarah's eye level} Total height=83.934 m+2.5 m\text{Total height} = 83.934 \text{ m} + 2.5 \text{ m} Total height=86.434 m\text{Total height} = 86.434 \text{ m}

    The height of the tree is approximately 86.43 m86.43 \text{ m}.

    86.43 m\boxed{\text{86.43 m}}
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