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From the top of a 20 m high cliff the angle of a depression of a boat is 30 degrees. Find the distance of the boat from the foot of the cliff.

Mathematics

ScanSolve AI Answer

Answer

020∘\text{}020^\circ\text{}

135∘\text{}135^\circ\text{}

47 feet\text{}47 \text{ feet}\text{}

Step-by-step solution

13 steps

5. From the top of a 20 m high cliff the angle of a depression of a b information on a diagram.

Since the question is cut off, we will draw a diagram showing a 20 m20 \text{ m} high cliff and a generic angle of depression.

  1. Draw a vertical line representing the cliff and a horizontal line from its top.

  2. Draw a line of sight from the top of the cliff downwards to an object.

  3. Label the height of the cliff as 20 m20 \text{ m} and the angle between the horizontal line of sight and the downward line of sight as the angle of depression, θ\theta.

    \begin{tikzpicture} % Cliff \draw[thick] (0,0) -- (0,2); \node[left] at (0,1) {$20 \text{ m}$}; \node[below] at (0,0) {Ground}; % Observer at top of cliff \node[above left] at (0,2) {Observer}; % Horizontal line of sight \draw[dashed] (0,2) -- (3,2); \node[above] at (1.5,2) {Horizontal Line of Sight}; % Object (boat) \node[below] at (3,0) {B}; \draw[thick] (0,0) -- (3,0); % Ground line % Line of sight to object \draw[thick] (0,2) -- (3,0); % Angle of depression \draw pic [draw, angle radius=0.7cm, "$\theta$"] {angle = { (3,2) -- (0,2) -- (3,0)}}; \end{tikzpicture}

    6. What is the back bearing of 200°?

  4. Determine if the given bearing is greater than or equal to 180∘180^\circ.

    The given bearing is 200∘200^\circ. Since 200∘≥180∘200^\circ \ge 180^\circ, we subtract 180∘180^\circ to find the back bearing.

  5. Calculate the back bearing.

    Back Bearing=200∘−180∘=20∘\text{Back Bearing} = 200^\circ - 180^\circ = 20^\circ

  6. Express the back bearing in three digits.

    The back bearing is 020∘\boxed{\text{}020^\circ\text{}}.

    7. A ship sails NW from a port to take supplies to an oil rig. What is the bearing to return from the oil rig to the port?

  7. Determine the bearing from the port to the oil rig.

    Sailing NW (North-West) means the direction is 45∘45^\circ west of North. Measured clockwise from North, this is 360∘−45∘=315∘360^\circ - 45^\circ = 315^\circ.
    So, the bearing from the port to the oil rig is 315∘315^\circ.

  8. Calculate the back bearing from the oil rig to the port.

    Since the bearing from the port to the oil rig (315∘315^\circ) is greater than 180∘180^\circ, we subtract 180∘180^\circ to find the back bearing.

    Back Bearing=315∘−180∘=135∘\text{Back Bearing} = 315^\circ - 180^\circ = 135^\circ

    The bearing to return from the oil rig to the port is 135∘\boxed{\text{}135^\circ\text{}}.

    8. Ben and Emma are out flying a kite. Emma can is making a 70° angle with the ground. The kite 50 feet away. To the nearest foot, how many f

    Assuming "50 feet away" refers to the length of the kite string (the hypotenuse of the right-angled triangle formed by the kite, the ground, and the string), and the question asks for the height of the kite.

  9. Identify the known values and the unknown.

    • Angle of elevation (θ\theta) = 70∘70^\circ
    • Length of kite string (hypotenuse) = 50 feet50 \text{ feet}
    • Height of the kite (opposite side) = hh

  10. Choose the appropriate trigonometric ratio.

    Since we know the hypotenuse and the angle, and we want to find the opposite side, we use the sine function:

    sin⁡(θ)=oppositehypotenuse\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}}

  11. Substitute the known values into the formula.

    sin⁡(70∘)=h50\sin(70^\circ) = \frac{h}{50}

  12. Solve for hh.

    h=50×sin⁡(70∘)h = 50 \times \sin(70^\circ)

  13. Calculate the value and round to the nearest foot.

    Using a calculator, sin⁡(70∘)≈0.93969\sin(70^\circ) \approx 0.93969.

    h≈50×0.93969h \approx 50 \times 0.93969 h≈46.9845h \approx 46.9845

    Rounding to the nearest foot:

    h≈47 feeth \approx 47 \text{ feet}

    The height of the kite is approximately 47 feet\boxed{\text{}47 \text{ feet}\text{}}.

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