The critical piece of information that Harrison has is the height of the helicopter above the ground. Knowing this height allows Harrison to determine the height of the well above the ground.
To explain this, we can consider the situation from a top-down view. Let's say that the well is at point A, the location of the catapult is at point C, and the helicopter is at point B. We know that the distance between B and C is 280 m, and the elevation angle at B is 15°.
By drawing a right-angled triangle with vertices at A, B, and C, we can use trigonometry to solve for the height of the well. The height of the well would be the height of the helicopter (which we don't know) minus the height of point B above the ground (which we can calculate).
If we let x be the height of the helicopter above the ground, and y be the height of point B above the ground, we can set up the following equation:
x - y = 125 m
To solve for y, we can use the trigonometric relationship:
tan(15°) = y / 280 m
Simplifying this equation, we have:
y = 280 m * tan(15°)
Now, by substituting this value of y into the first equation, we can solve for x:
x - 280 m * tan(15°) = 125 m
x = 280 m * tan(15°) + 125 m
Therefore, if Harrison knows the height of the helicopter above the ground, he can calculate the distance to the well by using the formula x = 280 m * tan(15°) + 125 m.
Harrison and Lara—two tireless adventurers who face mathematical roadblocks and puzzles wherever they go—have to use a catapult to fire a large rock over a hill and into a well that they can't see from the catapult's location. They are not permitted to walk anywhere on the hill, so measuring the distance to the well requires a clever plan.
Lara flies a helicopter over the well and drops a rope to measure her height above it. Exactly 125 m of rope is all she needs to touch the well; however, a steady wind is blowing the rope so it makes a straight line at an angle to the ground.
Meanwhile, back at the catapult, Harrison uses a rangefinder to determine that the helicopter is exactly 280 m away from the catapult, at an elevation of 15°.
Harrison radios Lara and confirms that the well, the helicopter and the catapult are all in the same vertical plane. Without making any further measurements, he now has enough information to set the distance for the catapult.
Harrison has one critical piece of information that you do not. Without it, you cannot determine the distance to the well with certainty. The information is not a measurement—Harrison will have no trouble determining it.
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