A lighthouse sits at the edge of a cliff, as shown. A ship at sea level is 750 meters from the base of the cliff. The angle of elevation from sea level to the base of the lighthouse is 24.7. The angle of elevation from sea level to the top of the lighthouse is 28.4 . Find the height of the lighthouse from the top of the cliff.

Do not round any intermediate computations. Round your answer to the nearest tenth.

Note that the figure below is not drawn to scale.

If the height is h, then

h = 750 tan28.4° - 750 tan24.7°

tan 28.4 = total height / 750

tan 24.7 = cliff height / 750
so
h = 750 ( tan 28.4 - tan 24.7)

Well, if the lighthouse were drawn to scale, it would probably look like a tiny toothpick compared to the real thing!

But let's put the jokes aside for a moment and solve this problem. From the information given, we have a right triangle formed by the ship, the base of the cliff, and the top of the lighthouse.

The tangent of an angle in a right triangle is equal to the ratio of the length of the side opposite the angle to the length of the side adjacent to the angle.

Using this, we can set up two equations:

tan(24.7) = height of the lighthouse / distance from the base of the cliff to the ship
tan(28.4) = height of the lighthouse + height of the cliff / distance from the base of the cliff to the ship

Now we have two equations with two unknowns: the height of the lighthouse and the height of the cliff. We can solve this system of equations to find the height of the lighthouse.

Let's call the height of the cliff "h" and the height of the lighthouse "L".

From the first equation, we have:

tan(24.7) = L / 750

Solving for L, we get:

L = tan(24.7) * 750

Now, let's substitute this value of L into the second equation:

tan(28.4) = (tan(24.7) * 750 + h) / 750

To solve for h, we multiply both sides by 750:

tan(28.4) * 750 = tan(24.7) * 750 + h

h = tan(28.4) * 750 - tan(24.7) * 750

Now we have the height of the cliff. To find the height of the lighthouse from the top of the cliff, we add the heights:

Height of lighthouse from the top of the cliff = L + h

I hope this doesn't make you lightheaded with all the calculations!

To find the height of the lighthouse from the top of the cliff, we can use trigonometric ratios and set up a triangle.

Let's assume that the height of the lighthouse from the top of the cliff is represented by 'h'.

We can create two right triangles using the given information:

1. The triangle formed by the ship, the base of the cliff, and the lighthouse's base. This triangle has an angle of elevation of 24.7 degrees.
2. The triangle formed by the ship, the top of the lighthouse, and the lighthouse's base. This triangle has an angle of elevation of 28.4 degrees.

From the first triangle, we can use the tangent function:

tan(24.7) = h / 750

Rearranging this equation, we can isolate 'h':

h = 750 * tan(24.7)

Now, we have the height of the lighthouse from the base of the cliff. However, we need to find the height from the top of the cliff, which can be found by subtracting the height of the cliff.

Let's assume the height of the cliff is represented by 'c'.

We can create a third right triangle formed by the lighthouse's base, the top of the lighthouse, and the top of the cliff. This triangle has a right angle.

From this triangle, we can use the tangent function again:

tan(28.4) = (h - c) / 750

Rearranging this equation, we can isolate '(h - c)':

(h - c) = 750 * tan(28.4)

Now, we have an equation for '(h - c)' in terms of the given information.

To find the height of the lighthouse from the top of the cliff, we need to substitute the values we know and calculate the result.

Using a calculator, evaluate the following expression:

(h - c) = 750 * tan(28.4)

The result will give you the height of the lighthouse from the top of the cliff. Remember to round your answer to the nearest tenth.