Question

A rescue helicopter hovering at an altitude of 3.5 km sights a campsite just over the peak of a mountain. Part A The horizontal distance of the helicopter from the mountain is 2.4 km. If the height of the mountain is 2.8 km, what is the horizontal distanced of the helicopter from the campsite? Explain Part B The groundspeed (horizontal speed} of the helicopter is 1.6 km/min. When will the helicopter reach the campsite? Explain. Part C The radio at the campsite can only transmit to a distance of 5 km. If the helicopter begins immediately to descend toward the campsite (along the diagonal line}, how far will the pilot be, horizontally, when he contacts the campsite?

Answer 1

The horizontal distance of the helicopter from the campsite is 0.1 km. It will take the helicopter 0.0625 minutes (approximately 3.75 seconds) to reach the campsite at a groundspeed of 1.6 km/min. The pilot will be 3.57 km away horizontally when he is able to contact the campsite within the 5 km transmission range.

Step-by-step explanation:

To solve for the horizontal distance of the helicopter from the campsite, we first need to recognize that we are dealing with a right-angled triangle. The peak of the mountain is the right angle, the horizontal distance from the helicopter to the peak of the mountain is the base of the triangle (2.4 km), and the difference in height between the helicopter's altitude and the mountain's peak (3.5 km - 2.8 km = 0.7 km) is the height of the triangle.

Using the Pythagorean theorem (a2 + b2 = c2), we calculate the hypotenuse, c, which represents the diagonal distance from the helicopter to the campsite, as follows:

c = √(2.42 + 0.72)

c = √(5.76 + 0.49)

c = √6.25

c = 2.5 km

To find the horizontal distance from the helicopter to the campsite, we subtract the known horizontal distance to the mountain peak:

Horizontal distance to the campsite = Hypotenuse - Horizontal distance to mountain

Horizontal distance to the campsite = 2.5 km - 2.4 km

Horizontal distance to the campsite = 0.1 km

For part B, given that the groundspeed of the helicopter is 1.6 km/min, we can calculate the time to reach the campsite by dividing the horizontal distance to the campsite by the groundspeed.

Time = Horizontal distance / Groundspeed

Time = 0.1 km / 1.6 km/min

Time = 0.0625 minutes

Finally, for part C, to calculate the horizontal distance from the campsite when the helicopter is within the 5 km transmission range of the campsite radio, we again have a right triangle with the altitude as one side (3.5 km) and the radio transmission range as the hypotenuse (5 km).

Using the Pythagorean theorem:

Horizontal distance when contacting2 = Transmission range2 - Altitude2

Horizontal distance when contacting2 = 52 - 3.52

Horizontal distance when contacting2 = 25 - 12.25

Horizontal distance when contacting = √(12.75)

Horizontal distance when contacting = 3.57 km

Therefore, the helicopter pilot will be 3.57 km away horizontally when he is able to contact the campsite.