A light house 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 light house is 40 feet relative to sea level. Which of the following correctly applies the sine ratio to find the distance from the light house operator on top of the light house to the ship? Round to the nearest hundredths place

Answer 1

To find the distance from the lighthouse operator to the ship using the sine ratio, we can use the following relationship in right triangles:

\[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \]

In this scenario:

• The angle of elevation (\(\theta\)) is 73 degrees.
• The height of the lighthouse (the opposite side) is 40 feet.
• The distance from the lighthouse operator to the ship (the hypotenuse) will be referred to as \(d\).

We can rearrange the sine formula to solve for \(d\):

\[ d = \frac{\text{opposite}}{\sin(\theta)}
\]

Substituting our values into this equation:

\[ d = \frac{40}{\sin(73^\circ)} \]

Now, we will calculate \(\sin(73^\circ)\):

\[ \sin(73^\circ) \approx 0.9563 \]

Now we can substitute this value back into the equation:

\[ d = \frac{40}{0.9563} \approx 41.85 \]

Thus, the distance from the lighthouse operator to the ship, rounded to the nearest hundredths place, is approximately 41.85 feet.

Answer 2

That’s wrong

Answer 3

Let’s go through the calculation again step by step to ensure accuracy.

1. Understanding the Triangle:

   • The height of the lighthouse (which is the opposite side) is 40 feet.
   • The angle of elevation from the ship to the lighthouse operator is 73 degrees.

2. Using the Sine Ratio: The sine of an angle in a right triangle is given by the formula: \[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \] Here, the "opposite" side is the height of the lighthouse (40 feet), and the "hypotenuse" is the distance from the lighthouse operator to the ship, which we want to find.

3. Rearranging the Formula: To find the hypotenuse \(d\): \[ d = \frac{\text{opposite}}{\sin(\theta)} = \frac{40}{\sin(73^\circ)} \]

4. Calculating \(\sin(73^\circ)\): Let's find \(\sin(73^\circ)\) using a calculator: \[ \sin(73^\circ) \approx 0.9563 \]

5. Calculating the Distance: Now we substitute this into our equation for \(d\): \[ d = \frac{40}{0.9563} \approx 41.85 \text{ feet} \]

Given the calculation above, the distance from the lighthouse operator to the ship remains approximately 41.85 feet. If you believe the answer still to be incorrect, please provide additional details or clarify any conditions related to the problem, and I would be happy to reassess it.