Miguel looks out from the crown of the Statue of Liberty approximately 250 ft above ground. He sights a ship coming into New York harbor and measures the angle of depression as 18 degrees. Find the distance from the base of the statue to the ship to the nearest foot.
We draw a rt. triangle:
A = 18 Deg.
X = Hor. side = Distance from base of statue to ship.
Y = 250 Ft = Ver. side.
tanA = Y/X
tan18 = 250/X
X = 250/tan18 = 769Ft.
Well, it seems like Miguel is really taking his sightseeing to new heights! Let's see if we can help him with some clown math.
We can start by visualizing the situation. Miguel is at the crown of the Statue of Liberty, which is 250 ft above the ground. He measures the angle of depression to the ship as 18 degrees.
Now, to find the distance from the base of the statue to the ship, we can use some trigonometry. The tangent of the angle of depression can help us here. The tangent of 18 degrees is equal to the opposite side (250 ft) divided by the distance we are looking for.
So, using some clown magic, we can write the equation as follows:
tan(18 degrees) = 250 ft / distance
To solve for distance, we can cross multiply:
distance * tan(18 degrees) = 250 ft
Now, let's unleash our clown calculator and find the value of distance:
distance = 250 ft / tan(18 degrees)
After some clown calculations, we find that the distance from the base of the statue to the ship is approximately 826 ft (to the nearest foot).
So, Miguel can wave a big hello to the ship about 826 feet away. Just make sure he doesn't drop any clown props when he waves!
To find the distance from the base of the statue to the ship, we can use the tangent function. The tangent of the angle of depression is equal to the ratio of the opposite side to the adjacent side.
In this case, the angle of depression is 18 degrees, and the opposite side is the height of the statue, which is 250 ft. The adjacent side is the distance from the base of the statue to the ship, which we want to find.
So, we have: tan(18) = 250/distance
To find the distance, we can rearrange the equation:
distance = 250 / tan(18)
Using a scientific calculator, we evaluate tan(18) to be approximately 0.32492. So, we have:
distance = 250 / 0.32492 ≈ 769.70 ft
Therefore, the distance from the base of the statue to the ship is approximately 769.70 feet.
To find the distance from the base of the statue to the ship, we can use trigonometry and the concept of an angle of depression.
Step 1: Draw a diagram to represent the given information. Label the following:
- The Statue of Liberty with a point labeled "S"
- The ship with a point labeled "P"
- The base of the statue with a point labeled "B"
- The angle of depression, marked as 18 degrees
Step 2: Identify the right-angled triangle formed by the points S, P, and B. The line SP represents the line of sight from the crown of the statue to the ship, and the line SB represents the vertical height from the crown to the base of the statue.
Step 3: Identify the trigonometric ratio that relates the angle of depression to the sides of the triangle. In this case, the tangent ratio will be helpful, as the tangent of an angle is equal to the ratio of the length of the opposite side to the length of the adjacent side.
Step 4: Apply the tangent ratio to find the length of the adjacent side, which is the distance from the base of the statue to the ship (BP). The tangent of the angle of depression (theta) is equal to the opposite side (SP) divided by the adjacent side (BP).
tan(theta) = SP / BP
Step 5: Solve the equation for BP. Rearrange the equation as follows:
BP = SP / tan(theta)
Step 6: Substitute the known values into the equation. In this case, theta = 18 degrees and SP = 250 ft.
BP = 250 ft / tan(18 degrees)
Step 7: Use a calculator to find the value of tan(18 degrees). Round the result to the nearest foot to get the final answer.
Once you have calculated the value, you will find that the distance from the base of the Statue of Liberty to the ship is approximately XX feet (rounded to the nearest foot).