the angle of elevation from point on the ground to the top of a pyramid is 20 degrees 20 degrees the angle elevation from a point 189 feet farther back to the top of the pyramid is 10 degrees 20 degrees. find the height of the pyramid
Let A be the top of the pyramid,
B = centre of pyramid at ground level
P=position of first observation (angle=20 deg)
Q=position of second observation (angle=10 degrees)
Then PQ=189 feet
and APQ is a triangle where
APQ=10°
AQP=180-20=160°
Consequently PAQ=10 deg by angles of a triangle.
Since triangle PAQ is an isosceles triangle, we deduce that PA=189 feet.
and the height of the pyramid is
h=189sin(20 deg).
Note:
This solutions takes advantage of the fact that the triangle is isosceles.
In general, if it is not, the triangle can be solved using the sine rule, and h found consequently.
If you have not done sine rule already, then you can let h=AB, and x=PB.
Form two equations using each of the triangles APB and AQB in terms of h and x. Solve for h by eliminating x.
To find the height of the pyramid, we can use basic trigonometry. Let's start by drawing a diagram to visualize the problem.
We have a pyramid, and there are two points on the ground. Point A is the closer point, and the angle of elevation from this point to the top of the pyramid is 20 degrees. Point B is the point 189 feet further back, and the angle of elevation from this point to the top of the pyramid is 10 degrees.
Let's denote the height of the pyramid as "h".
Now, let's consider triangle ABC, where A is the point on the ground closest to the pyramid, B is the point 189 feet further back, and C is the top of the pyramid.
We know that the angle of elevation from A to C is 20 degrees, and the angle of elevation from B to C is 10 degrees.
Using trigonometry, we can write two equations:
1. tan(20°) = h / x (Equation 1)
Where x is the distance between point A and the base of the pyramid.
2. tan(10°) = h / (x + 189) (Equation 2)
Where (x + 189) is the distance between point B and the base of the pyramid.
We have two equations and two unknowns (h and x). Now we can solve this system of equations to find the height of the pyramid.
Rearrange Equation 1 to solve for x:
x = h / tan(20°)
Substitute this value into Equation 2:
tan(10°) = h / (h / tan(20°) + 189)
Now we can solve this equation for h. By rearranging and simplifying, we get:
h = (tan(10°) * tan(20°) * 189) / (tan(20°) - tan(10°))
Calculating this expression will give us the height of the pyramid.