A lighthouse operator is sending signals to a ship. the angle of elevation from the ship to the lighthouse operator is 73 degrees. the height of the lighthouse is 40 feet relative to sea level. which of the following correctly applies the sine ratio to find the distance from the lighthouse operator on top of the light house to the ship? round to the nearest hundreth place.

Answer 1

To find the distance from the lighthouse operator at the top of the lighthouse to the ship using the sine ratio, we need to set up the right triangle formed by the height of the lighthouse, the distance from the ship to the base of the lighthouse, and the line of sight from the lighthouse operator to the ship.

1. Identify the triangle: Let's denote:

   • \( h = \) height of the lighthouse = 40 feet
   • \( \theta = 73^\circ \) (angle of elevation from the ship to the lighthouse)
   • \( d = \) distance from the ship to the base of the lighthouse (unknown)
   • \( x = \) distance from the ship to the lighthouse operator (hypotenuse)

2. Using the sine function: The sine ratio is defined as: \[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{h}{x} \] where:

   • the opposite side is the height of the lighthouse (40 feet),
   • the hypotenuse is the distance from the ship to the lighthouse operator.

3. Set up the equation: \[ \sin(73^\circ) = \frac{40}{x} \]

4. Solve for \( x \): Rearranging the equation gives: \[ x = \frac{40}{\sin(73^\circ)} \]

5. Calculate \( x \): First, find \( \sin(73^\circ) \). Using a calculator, \[ \sin(73^\circ) \approx 0.9563 \]

   Now plug this value into the equation: \[ x = \frac{40}{0.9563} \approx 41.85 \text{ feet} \]

Thus, the distance from the lighthouse operator to the ship is approximately 41.85 feet.

Answer 2

this is wrong

Answer 3

I apologize for the misunderstanding. Let’s clarify the problem step by step to make sure we find the correct distance.

We have:

• Height of the lighthouse (h): 40 feet
• Angle of elevation (θ): 73 degrees

To find the distance (d) from the base of the lighthouse to the ship (the opposite side in our right triangle setup), we use the tangent function instead of sine, since we have the height and are interested in the horizontal distance from the ship to the base of the lighthouse.

Here's how you set it up:

1. Using the tangent function: \[ \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} = \frac{h}{d} \] Thus, \[ d = \frac{h}{\tan(\theta)} \]

2. Substituting the values: \[ d = \frac{40}{\tan(73^\circ)} \]

3. Calculating \(\tan(73^\circ)\): Using a calculator, \[ \tan(73^\circ) \approx 3.4874 \]

4. Now, substituting this value back: \[ d = \frac{40}{3.4874} \approx 11.47 \text{ feet} \]

This distance, \( d \), is the horizontal distance from the base of the lighthouse to the ship.

Since you need the distance from the lighthouse operator at the top of the lighthouse to the ship (the hypotenuse), we can now use the Pythagorean theorem.

Hypotenuse Calculation

Using:

• \( h = 40 \) feet (height)
• \( d \approx 11.47 \) feet (base distance)

The hypotenuse (x) can be calculated as: \[ x = \sqrt{h^2 + d^2} \] \[ x = \sqrt{(40)^2 + (11.47)^2} = \sqrt{1600 + 131.49} = \sqrt{1731.49} \approx 41.6 \text{ feet} \]

Thus, the distance from the lighthouse operator to the ship is approximately 41.6 feet.

Thank you for your patience, and I hope this clears up the issue!