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 the figure. let h height of momnument. Let d be the distance from the bottom of the tower to the bottom of the monument.
Draw a dotted horizonal plane from the top of the monument to the tower. Let the distance from this line to the top of the tower be 145-h
from the lower triangle, d/145=sin(90-31) solve for d.
Now knowing d, look a the top triangle (base dotted line
d/(145-h)=sin((90-13) you know d, so solve for h.
To find the height of the monument, we can use the trigonometric concept of angles of depression.
Let's consider the triangle formed by the tower, the top of the monument, and the bottom of the monument.
Let H be the height of the monument.
In the triangle formed:
From the top of the tower to the top of the monument, the angle of depression is 13 degrees.
From the top of the tower to the bottom of the monument, the angle of depression is 31 degrees.
The height of the tower is given to be 145 ft.
Let's label the relevant parts of the triangle:
A = Angle of depression from the top of the tower to the top of the monument (13 degrees)
B = Angle of depression from the top of the tower to the bottom of the monument (31 degrees)
C = Right angle (90 degrees)
D = Top of the tower
E = Top of the monument
F = Bottom of the monument
G = Bottom of the tower
Using trigonometry, we can write the following equation for tangent:
tan(A) = H / x (1)
tan(B) = (H + 145) / x (2)
where x represents the horizontal distance from the top of the tower to the monument.
Rearranging equation (1), we have:
x = H / tan(A) (3)
Substituting equation (3) into equation (2), we get:
tan(B) = (H + 145) / (H / tan(A))
tan(B) = (H + 145) tan(A) / H
Simplifying further:
tan(B) = tan(A) + 145 tan(A) / H
tan(B) - tan(A) = 145 tan(A) / H
Using the trigonometric identity tan(B) - tan(A) = tan(B - A), we can rewrite the equation:
tan(B - A) = 145 tan(A) / H
Now, we solve for H:
H = 145 tan(A) / tan(B - A)
Plugging in the given values for A and B:
H = 145 tan(13) / tan(31 - 13) ≈ 145 (0.224) / tan(18) ≈ 145 (0.224) / (0.3249) ≈ 99.3 ft
Therefore, the height of the monument is approximately 99.3 ft.
To find the height of the monument, we can use trigonometry and the concept of angles of depression.
Let's break down the problem and label the given information:
1. The angle of depression from the top of the tower to the top of the monument is 13 degrees.
2. The angle of depression from the top of the tower to the bottom of the monument is 31 degrees.
3. The height of the tower is 145 ft.
To visualize this, draw a diagram with a horizontal line representing the ground, a vertical line representing the tower, and another vertical line representing the monument.
First, let's find the distance from the top of the tower to the bottom of the monument. We'll call this distance x.
Using trigonometry, we can set up the following equation:
tan(31 degrees) = height of the monument / x
Rearranging the equation, we get:
x = height of the monument / tan(31 degrees)
Next, let's find the distance from the top of the tower to the top of the monument, which is x plus the height of the monument. We'll call this distance y.
Using trigonometry again, we can set up the following equation:
tan(13 degrees) = height of the monument / y
Rearranging the equation, we get:
y = height of the monument / tan(13 degrees)
Now, we can substitute the value of y into x + height of the monument, since y = x + height of the monument:
x + height of the monument = height of the monument / tan(13 degrees)
Now, let's solve this equation for the height of the monument:
x = height of the monument / tan(31 degrees)
x + height of the monument = height of the monument / tan(13 degrees)
We can substitute the previously derived value of x in the second equation:
height of the monument / tan(31 degrees) + height of the monument = height of the monument / tan(13 degrees)
To simplify the equation, we can multiply both sides by tan(13 degrees) to eliminate the denominators:
height of the monument + height of the monument * (tan(31 degrees) / tan(13 degrees)) = height of the monument
Simplifying further:
height of the monument * (1 + tan(31 degrees) / tan(13 degrees)) = height of the monument
Now, divide both sides by (1 + tan(31 degrees) / tan(13 degrees)):
height of the monument = height of the monument / (1 + tan(31 degrees) / tan(13 degrees))
Finally, we can solve for the height of the monument. Substituting the given values into the equation:
height of the monument = 145 ft / (1 + tan(31 degrees) / tan(13 degrees))
Calculating this, the height of the monument is approximately 72.61 ft.