A TOWER AND A MONUMENT STAND ON A LEVEL PLANE.THE ANGLES OF DEPRESSION OF THE TOP AND BOTTOM OF THE MONUMENT VIEWED FROM THE TOP OF THE TOWER ARE 13DEGREE AND 31 DEGREE,RESPECTIVELY;THE HEIGHT OF THE TOWER IS 145 FT. FIND THE HEIGHT OF THE MONUMENT.

Draw a diagram. Let

T be the top of the tower
M be the top of the monument
A be the base of the tower
B be the base of the monument

∠TBA = 31°
so, 145/TB = sin 31°

Since
∠TBM = 59°
∠MTB = 18°,
∠TMB = 103°

The desired height (MB) is found by

MB/sin18° = TB/sin103°

To find the height of the monument, we can use the concept of trigonometry.

Let's assume the height of the monument is "h" feet.

From the top of the tower, the angle of depression to the top of the monument is 13 degrees. This means that if we draw a line from the top of the tower to the top of the monument, it will form an angle of 13 degrees with the horizontal level.

Similarly, from the top of the tower, the angle of depression to the bottom of the monument is 31 degrees. This means that if we draw a line from the top of the tower to the bottom of the monument, it will form an angle of 31 degrees with the horizontal level.

Using trigonometry, we can relate the height of the monument to the height of the tower.

Let's consider the right-angled triangle formed by the tower, the top of the monument, and the bottom of the monument.

In this triangle, the side opposite the angle of depression of 31 degrees is the height of the monument (h), the side opposite the angle of depression of 13 degrees is the height of the tower (145 ft), and the hypotenuse is the distance between the tower and the monument.

Since we have two sides and the included angle, we can use the tangent function to find the height of the monument.

Using the formula for tangent:

tangent(angle) = opposite / adjacent

tangent(31 degrees) = h / 145 ft

Therefore, h = tangent(31 degrees) * 145 ft

Calculating this expression, we find:

h ≈ 81.219 ft

So, the height of the monument is approximately 81.219 feet.

To find the height of the monument, we can use the concept of trigonometry.

Let's denote:
- h as the height of the monument (which we want to find)
- d as the horizontal distance between the tower and the monument

We are given the angles of depression from the top of the tower to the top and bottom of the monument. The angle of depression is defined as the angle formed between a line of sight from the observer (in this case, the top of the tower) to a point below the horizontal line (the bottom of the monument in this case).

Since the tower and the monument are on a level plane, we can consider a triangle formed by the tower, the bottom of the monument, and the top of the monument. This is a right triangle.

Now, let's consider the angle of depression to the top of the monument. In this case, the opposite side is the height of the tower (145 ft), and the adjacent side is the horizontal distance between the tower and the monument (d).

Using the tangent function, we can write:

tan(13 degrees) = (h + 145) / d

Next, let's consider the angle of depression to the bottom of the monument. In this case, the opposite side is the height of the tower (145 ft + h, since we now include the height of the monument), and the adjacent side is still the horizontal distance between the tower and the monument (d).

Using the tangent function again, we can write:

tan(31 degrees) = (h + 145) / d

Now we have a system of two equations with two variables. By solving these equations simultaneously, we can find the values of h and d.

Let's rearrange the equations to solve for d in terms of h:

d = (h + 145) / tan(13 degrees)

d = (h + 145) / tan(31 degrees)

Now, equate the expressions for d:

(h + 145) / tan(13 degrees) = (h + 145) / tan(31 degrees)

Cross-multiply to get rid of the denominators:

tan(31 degrees) * (h + 145) = tan(13 degrees) * (h + 145)

Simplify:

tan(31 degrees)*h + tan(31 degrees)*145 = tan(13 degrees)*h + tan(13 degrees)*145

Rearrange the terms:

tan(31 degrees)*h - tan(13 degrees)*h = tan(13 degrees)*145 - tan(31 degrees)*145

Factor out h:

h * (tan(31 degrees) - tan(13 degrees)) = tan(13 degrees)*145 - tan(31 degrees)*145

Divide both sides by (tan(31 degrees) - tan(13 degrees)):

h = (tan(13 degrees)*145 - tan(31 degrees)*145) / (tan(31 degrees) - tan(13 degrees))

Now, we can plug in the values of tan(13 degrees), tan(31 degrees), and the height of the tower (145 ft) to calculate the height of the monument (h).