a person who eyes are 5 feet above the ground observes the top of a building to have an angle of elevation of 40 degree. The person walks 100 feet closer to the building and observes that the top of the building now has an angle of elevation of 65 degree. How high is the building?

Let the height of the building be h; let the man start at distance x from the building.

(h-5)/x = tan40 = 0.839
(h-5)/(x-100) = tan60 = 1.732

h = 167.7 ft
x = 193.9 ft

check:
162.7/193.9 = 0.839
162.7/93.9 = 1.732

To find the height of the building, we can use trigonometry and set up a proportion based on the angle of elevation.

Let's assign some variables:
Let h be the height of the building.
Let d1 be the distance from the initial position of the person to the building.
Let d2 be the distance from the new position of the person to the building.

We know that the person's eyes are 5 feet above the ground, so at the initial position, the person's line of sight is parallel to the ground. This means that the height of the building is the same as the distance from the initial position of the person to the top of the building.

Using trigonometry, we can set up the following proportions:

tan(40°) = h / d1 (Equation 1)
tan(65°) = h / d2 (Equation 2)

Now, we need to solve these equations to find the value of h.

First, let's solve Equation 1 for h:
h = tan(40°) * d1

Next, let's solve Equation 2 for h:
h = tan(65°) * d2

Since h represents the same value in both equations, we can set the two expressions for h equal to each other and solve for d1:

tan(40°) * d1 = tan(65°) * d2

Now, we can solve for d1:
d1 = (tan(65°) * d2) / tan(40°)

Since we know that the person walks 100 feet closer to the building, we can substitute d1 with (d2 - 100) in the equation:

(d2 - 100) = (tan(65°) * d2) / tan(40°)

Re-arranging the equation to solve for d2:

tan(40°) * (d2 - 100) = tan(65°) * d2

Expanding the equation:

tan(40°) * d2 - 100 * tan(40°) = tan(65°) * d2

Transferring all terms involving d2 to one side:

tan(40°) * d2 - tan(65°) * d2 = 100 * tan(40°)

Calculating the value:

d2 * (tan(40°) - tan(65°)) = 100 * tan(40°)

Finally, solving for d2:

d2 = (100 * tan(40°)) / (tan(40°) - tan(65°))

Now that we know the value of d2, we can substitute it back into Equation 2 to find the height of the building (h):

h = tan(65°) * d2

Calculating the value:

h = tan(65°) * [(100 * tan(40°)) / (tan(40°) - tan(65°))]

Therefore, the height of the building is given by the calculated value of h using the above expression.