An antenna stands on a pole on the top of a house.If a woman is 70m from the house observes that the angles of elevation of the top and foot of the pole are 63 and 60 resp.,how long is the pole?
Draw the diagram.
If the height is h, note that
h = 70tan63° - 70tan60°
To find the length of the pole, we can use trigonometry and the concept of right triangles. Let's label the height of the pole as h and the distance from the woman to the foot of the pole as x.
We have two right triangles here: the triangle formed by the height of the pole, the distance from the woman to the top of the pole, and the line connecting these two points, and the triangle formed by the height of the pole, the distance from the woman to the foot of the pole, and the line connecting these two points.
First, let's consider the triangle formed by the height of the pole, the distance from the woman to the top of the pole, and the line connecting these two points. We have the angle of elevation of the top of the pole as 63 degrees, which means that the tangent of this angle is equal to the ratio of the height of the pole to the distance from the woman to the top of the pole. In mathematical terms:
tan(63) = h / (x + 70)
Next, let's consider the triangle formed by the height of the pole, the distance from the woman to the foot of the pole, and the line connecting these two points. We have the angle of elevation of the foot of the pole as 60 degrees, which means that the tangent of this angle is equal to the ratio of the height of the pole to the distance from the woman to the foot of the pole. In mathematical terms:
tan(60) = h / x
Now, we have two equations with two variables (h and x), which we can solve simultaneously to find the values of h and x.
By rearranging the second equation, we get:
h = x * tan(60)
Substituting this value of h in the first equation, we have:
tan(63) = (x * tan(60)) / (x + 70)
Simplifying further, we can solve this equation to find the value of x:
x = (tan(63) * (x + 70)) / tan(60)
Now, we can solve this equation using numerical methods or by approximate calculations to find the value of x. Once we have the value of x, we can substitute it in the equation h = x * tan(60) to find the height of the pole. Finally, we can calculate the length of the pole by adding the height of the pole to 70m (the distance from the woman to the foot of the pole).
Please note that this explanation outlines the mathematical approach to solving the problem. The actual numerical calculations may require a calculator or mathematical software to obtain the precise values.