From a ship, the angle of elevation of the top of the lighthouse is 30 45. What is the horizontal distance of the ship from the lighthouse is 42 meters tall?
tan 30.75° = 42/x , x is the horizontal distance
x = 42/tan30.75 = ....
To calculate the horizontal distance of the ship from the lighthouse, we can use trigonometry. Let's assume that the height of the ship's eye level from the ground is "h" meters.
First, let's draw a diagram to visualize the problem:
Lighthouse
|\
| \
| \
h| \t
| \
| \
|______\
d Ship
In this diagram, the angle of elevation from the ship to the top of the lighthouse is 30°, and the height of the lighthouse is 42 meters. The horizontal distance between the ship and the lighthouse is represented by "d".
Now, we can use the tangent function to solve for "d":
tan(angle) = opposite/adjacent
In this case, the opposite side is the height of the lighthouse (42 meters) and the adjacent side is the horizontal distance "d" that we want to find.
So we have:
tan(30°) = 42 / d
To find the value of "d", we need to rearrange the equation:
d = 42 / tan(30°)
Now we can calculate "d" using a calculator:
d ≈ 42 / 0.5774
d ≈ 72.71 meters
Therefore, the horizontal distance of the ship from the lighthouse is approximately 72.71 meters.
To find the horizontal distance between the ship and the lighthouse, we can use trigonometry.
Let's label the height of the lighthouse as 'h' (which is given as 42 meters) and the angle of elevation as 'θ' (which is given as 30°).
Using the trigonometric function tangent (tanθ = opposite/adjacent), we can find the horizontal distance between the ship and the lighthouse. In this case, the opposite side is the height of the lighthouse, and the adjacent side is the horizontal distance we want to find.
The formula to find the adjacent side is:
adjacent = opposite / tanθ
Substituting the given values into the formula:
adjacent = 42 meters / tan 30°
Now we need to calculate the tangent of 30°.
You can use a scientific calculator, and many calculators have a "tan" or "tan^-1" function which allows you to find the tangent of an angle.
If you don't have a calculator, you can use a table of trigonometric values. Look up the tangent for 30°, and it should be √3 / 3.
Now, substitute this value back into the formula:
adjacent = 42 meters / (√3 / 3)
To simplify the equation, multiply the numerator and denominator by the reciprocal of the square root of 3. The reciprocal of a number is 1 divided by the number.
adjacent = 42 meters * (3 / √3)
To rationalize the denominator, multiply the numerator and denominator by the square root of 3:
adjacent = 42 meters * (3 / (√3 * √3))
This simplifies to:
adjacent = 42 meters * 3√3 / 3
Now, cancel out the 3's:
adjacent = 42 meters * √3
To find the numerical answer, calculate the product:
adjacent = 42 meters * √3 ≈ 72.75 meters
Therefore, the horizontal distance of the ship from the lighthouse is approximately 72.75 meters.