from a point P on a level ground and directly west of a pole,the angle of elevation from the top of the pole is 045 and from point Q east of the pole the angle of elevation of the pole the angle of elevation of the top of the pole is 068 if /PQ/ is 10m,what is the distance from P to the pole and the height of the pole
Draw a diagram. Review your basic trig functions. If the height of the pole is h, then
h cot45° + h cot68° = 10
To find the distance from point P to the pole, we can use trigonometry. Let's break down the problem step by step:
1. From point P, the angle of elevation to the top of the pole is 045. This means that if we draw a straight line from P to the top of the pole, it will form a right angle with the ground.
2. Similarly, from point Q, the angle of elevation to the top of the pole is 068. Again, this implies that if we draw a straight line from Q to the top of the pole, it will form a right angle with the ground.
3. We know that the distance between point P and Q, which is represented as /PQ/, is 10 meters.
Let's label the distance from P to the pole as "x" and the height of the pole as "h."
Now, we can use the tangent function to find the values of x and h.
Tangent of an angle is defined as the ratio of the opposite side to the adjacent side.
From point P, tangent(045) = h / x (1)
From point Q, tangent(068) = h / (10 + x) (2)
Since we have two equations and two unknowns (x and h), we can solve them simultaneously.
Dividing equation (1) by equation (2), we get:
[tan(045)] / [tan(068)] = h / x / [h / (10 + x)]
Simplifying the above equation gives us:
tan(045) * (10 + x) = tan(068) * x
Now, we can substitute the values of tangent(045), tangent(068), and solve for x:
(1) * (10 + x) = (2) * x
tan(045) * (10 + x) = tan(068) * x
0.9999 * (10 + x) = 2.7372 * x
9.999 + 0.9999 * x = 2.7372 * x
0.9999 * x - 2.7372 * x = -9.999
-1.7373 * x = -9.999
x = -9.999 / -1.7373
x ≈ 5.756 meters
So, the distance from point P to the pole is approximately 5.756 meters.
To find the height of the pole, we can substitute this value of x into either equation (1) or (2):
From equation (1):
tangent(045) = h / x
0.9999 = h / 5.756
h = 0.9999 * 5.756
h ≈ 5.754 meters
Therefore, the height of the pole is approximately 5.754 meters.