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.
What information does Harrison have that you do not? What are the possible distances to the well? By the end of this module, you should know how to solve this riddle.
The critical piece of information that Harrison has is the height of the helicopter above the ground. With this information, he can use trigonometry to determine the distance to the well with certainty.
To solve this riddle, we can use the concept of similar triangles. Let's assume that the height of the helicopter above the ground is h meters.
Using the given information, we know that the helicopter is 280 meters away from the catapult at an elevation of 15°. This forms a right triangle, where the opposite side is h and the adjacent side is 280 meters. Therefore, we can use the tangent function to find the value of h:
tan(15°) = h/280
Solving for h:
h = 280 * tan(15°)
Now, let's consider the triangle formed by the helicopter, the well, and the point where the rope touches the ground. This triangle is also a right triangle, where the opposite side is h and the hypotenuse is 125 meters (the length of the rope). We can use the sine function to relate the angle between the hypotenuse and the opposite side:
sin(theta) = h/125
Again, solving for h:
h = 125 * sin(theta)
Since the triangles are similar, we can equate the two expressions for h:
280 * tan(15°) = 125 * sin(theta)
We can now solve this equation to find the value of theta, which represents the angle between the hypotenuse and the opposite side in the second triangle. Once we have the value of theta, we can use the sine function again to find the distance to the well:
sin(theta) = h/125
distance to well = 125 / sin(theta)
Therefore, the possible distances to the well depend on the value of theta, which we can find by solving the equation 280 * tan(15°) = 125 * sin(theta).
To solve this riddle, we need to use trigonometry. Let's analyze the information given:
1. Lara drops a rope of 125 m from the helicopter to touch the well.
2. The rope makes a straight line at an angle to the ground.
3. Harrison uses a rangefinder to determine that the helicopter is 280 m away from the catapult, at an elevation of 15°.
4. The well, the helicopter, and the catapult are in the same vertical plane.
To determine the missing information that Harrison has, we need to consider the concept of vertical angles. When two lines intersect, the opposite angles formed are called vertical angles, and they are equal.
In this case, the vertical angle formed between the line connecting the helicopter and the catapult and the line connecting the helicopter and the well is equal to the angle of elevation of 15°. This means that the angle between the rope and the ground is also 15°.
Now, we can solve the problem using trigonometry. Let's define the variables:
x = distance from the helicopter to the well (the missing information)
y = distance from the helicopter to the point where the rope touches the ground
Using trigonometry, we can set up the following equations:
y = x * tan(15°) (1)
280 - y = x (2)
From equation (1), we can solve for y:
y = x * tan(15°)
Substituting this into equation (2), we get:
280 - x * tan(15°) = x
Simplifying the equation, we have:
280 = x + x * tan(15°)
280 = x(1 + tan(15°))
Dividing both sides by (1 + tan(15°)), we find:
x = 280 / (1 + tan(15°))
Now, we can calculate the possible values of x by substituting the known angle of 15° into the equation:
x ≈ 254.55 meters
Therefore, the distance to the well is approximately 254.55 meters.