To measure a stone face carved on the side of a mountain, two sightings 407 feet from the base of the mountain are taken. If the angle of elevation to the bottom of the face is 36 degrees and the angle of elevation to the top of the face is 67 degrees, what is the height of the stone face?

review the definition of tan(x).

Draw a diagram. You will see that the height is

407 tan67° - 407 tan36°

To find the height of the stone face, we can use trigonometry. Let's consider a right triangle formed by the base of the mountain, the bottom of the stone face, and a line connecting the top of the stone face to the base.

First, let's find the length of the base of the right triangle. We have two sightings taken 407 feet from the base of the mountain, forming two right triangles. Since we are only interested in the length of the base, we can ignore the height component of the triangle.

Using trigonometry, we can write the equation:
tan(angle of elevation) = height / base

For the bottom of the stone face:
tan(36 degrees) = height / base
base = height / tan(36 degrees)

For the top of the stone face:
tan(67 degrees) = height / base
base = height / tan(67 degrees)

Since both bases are equal, we can set the two equations equal to each other:
height / tan(36 degrees) = height / tan(67 degrees)

Now, we can solve for the height:
height = base * tan(36 degrees)
height = 407 feet * tan(36 degrees)
height ≈ 275.42 feet

Therefore, the height of the stone face is approximately 275.42 feet.

To find the height of the stone face, we can use the trigonometric concept of tangent.

Let's denote the height of the stone face as "x".

From the given information, we have two right-angled triangles formed. One triangle represents the bottom half of the stone face, and the other triangle represents the top half of the stone face.

Using the trigonometric ratio tangent, we can write:

tan(36 degrees) = (height of bottom half) / (407 feet)
tan(67 degrees) = (height of top half) / (407 feet)

We can rewrite these equations as:

x / 407 = tan(36 degrees)
x / 407 = tan(67 degrees)

To find the height of the entire stone face, we need to add the height of the bottom half and the height of the top half:

Height of stone face = (height of bottom half) + (height of top half)
= 407 * tan(36 degrees) + 407 * tan(67 degrees)

Now, let's calculate the values:

Height of stone face = 407 * tan(36 degrees) + 407 * tan(67 degrees)
≈ 407 * 0.7265 + 407 * 2.737
≈ 295.8 + 1114.6
≈ 1410.4 feet

Therefore, the height of the stone face is approximately 1410.4 feet.