... a straight road observes that
at two points l km apart, the
angles of elevation of a pole in
front of him are 30° and 75°.
It would help if you proofread your questions before you posted them.
"a straight road observes" nothing.
What is your question?
250(2.732)
To solve this problem, we can use the concept of trigonometry. Let's assume that the person is standing at point A which is l km away from the pole, and the first observation is at point B with an angle of elevation of 30°. The second observation is at point C with an angle of elevation of 75°.
Now, let's break down the problem step by step:
Step 1: Draw a diagram
Start by drawing a straight line to represent the road, and mark points A, B, and C accordingly.
A---------B-----C-----------Pole
|-----------------l km--------------|
Step 2: Calculate the height of the pole
Let's allocate a variable 'h' to represent the height of the pole. We need to find the value of 'h'.
Step 3: Use trigonometry to find the height of the pole
From point B, the angle of elevation of 30° can be associated with the tangent function:
tan(30°) = h / l
Similarly, from point C, the angle of elevation of 75° can be associated with the tangent function:
tan(75°) = h / (l + l)
Step 4: Solve the equations
Now, we have two equations in two unknowns:
tan(30°) = h / l
tan(75°) = h / (2l)
We can solve these equations simultaneously to find the value of h.
Step 5: Substitute the values and solve
Using a scientific calculator or trigonometric tables, we can find the values of tan(30°) and tan(75°). Let's assume they are equal to x and y, respectively.
x = tan(30°)
y = tan(75°)
Now we can substitute these values into the equations:
x = h / l
y = h / (2l)
Solving for h, we get:
h = x * l
h = y * (2l)
Therefore, h = (x * l) = (y * 2l).
Step 6: Calculate the values
Finally, substitute the values of l, x, and y into the equation h = (x * l) = (y * 2l) to find the height of the pole.
I hope this step-by-step explanation helps you understand how to solve problems involving angles of elevation.