Casey sights the top of an 84 foot tall lighthouse at an angle of elevation of 58 degrees. If Casey is 6 feet tall, how far is he standing from the base of the lighthouse?

tan 58 = (84-6) / x

To solve this problem, we can use trigonometry, specifically the tangent function, which relates the angle of elevation, the opposite side, and the adjacent side.

Let's call the distance Casey is standing from the base of the lighthouse "x."

We know that the height of the lighthouse is 84 feet, and Casey's height is 6 feet.

Using the concept of similar triangles, we can set up the following proportion:

tan(58 degrees) = opposite side / adjacent side
tan(58 degrees) = (84 - 6) / x
tan(58 degrees) = 78 / x

We can now solve for x by cross-multiplying and then dividing:

x * tan(58 degrees) = 78
x = 78 / tan(58 degrees)

Calculating the value of x, we have:

x ≈ 57.5 feet

Therefore, Casey is standing approximately 57.5 feet away from the base of the lighthouse.

To find the distance Casey is standing from the base of the lighthouse, we can use trigonometry.

First, let's draw a right triangle. The height of the lighthouse forms one side of the triangle, and the distance from Casey to the lighthouse forms another side. The angle of elevation, 58 degrees, is formed by the height of the lighthouse and the line of sight from Casey's eye to the top of the lighthouse.

We can label the sides of the triangle as follows:
- The side opposite the angle of elevation is the height of the lighthouse, 84 feet.
- The side adjacent to the angle of elevation is the distance from Casey to the lighthouse (let's call it x).
- The hypotenuse of the triangle is the line of sight from Casey's eye to the top of the lighthouse.

Now, we can use the tangent function to set up an equation:
tan(58 degrees) = opposite / adjacent.
tan(58 degrees) = 84 / x.

To solve for x, we can rearrange the equation:
x = 84 / tan(58 degrees).

Plugging the values into a calculator, we get:
x ≈ 84 / 1.583.
x ≈ 53.032 feet.

Therefore, Casey is standing approximately 53.032 feet from the base of the lighthouse.