a flagpole is 25 feet tall. a truck runs into the pole and bends it at the very bottom of the pole but the rest of the pole remains straight. after the accident we measure the angle of elevation to the top of the pole from a point 30 feet from the base of the pole in the same direction as the pole bends. the angle of elevation is 45 degrees. what is the distance between where the top of the pole used to be and where it ended up?

Question

a flagpole is 25 feet tall. a truck runs into the pole and bends it at the very bottom of the pole but the rest of the pole remains straight. after the accident we measure the angle of elevation to the top of the pole from a point 30 feet from the base of the pole in the same direction as the pole bends. the angle of elevation is 45 degrees. what is the distance between where the top of the pole used to be and where it ended up?

Answer 1

If the top of the pole moved from P to Q, and the base of the pole is at A, and the observation point is at B,

sinQ/30 = sin45/25
sinQ = 0.848
Q = 58°

Now you have two angles of triangle ABQ, so you know angle θ=QAB.

PQ^2 = 25^2 + 25^2 - 2(25)(25)cos(90-θ)
= 1250 - 1250sinθ
= 1250(1-sin(pi-(45+Q))
= 1250(1-sin(45°+58°))
= 32
PQ = √32 = 5.66

Answer 2

To find the distance between where the top of the pole used to be and where it ended up, we need to find the height of the new bent portion of the pole.

Let's solve this step-by-step:

1. Let's draw a diagram to visualize the situation. Mark the original position of the top of the pole and the point where the truck hit the pole at the bottom. Draw a straight line connecting these two points to represent the original pole. Label the height of the pole as 25 feet.

2. From the point where the truck hit the pole (the bottom), draw a straight line perpendicular to the ground. This represents the bent portion of the pole.

3. Measure the distance from the base of the pole to the point where you are standing (the observer) at a right angle from the bend and mark it as 30 feet.

4. Measure the angle of elevation from your position to the top of the pole (after the bend), which is given as 45 degrees.

Now, we can use trigonometry to find the height of the new bent portion of the pole.

5. Based on the information given, we have a right-angled triangle formed by the bent portion of the pole, the distance from the bend to the observer (30 feet), and the angle of elevation (45 degrees).

6. We can use the tangent function to relate the angle and the opposite side (the height of the bent portion of the pole). The formula for tangent is: tan(angle) = opposite/adjacent.

In this case, the opposite side is the height of the pole and the adjacent side is the distance from the bend to the observer.

So, tan(45 degrees) = height of pole/30 feet.

7. Rearranging the equation, height of pole = tan(45 degrees) * 30 feet.

Calculating this, we find that the height of the bent portion of the pole is approximately 30 feet.

Therefore, the distance between where the top of the pole used to be and where it ended up is the same as the height of the bent portion of the pole, which is 30 feet.