Question

A cable car travels 2236 m to climb a

mountain that is 1000 m high. It then
travels 1414 m to descend 1000 m on the
other side of the mountain. A tunnel goes
straight through the mountain, from the
starting point to the endpoint of the cable
car’s journey. How long is the tunnel?

Answer 1

To determine the length of the tunnel through the mountain, we can apply the Pythagorean theorem to the right triangle formed by the mountain's height and the cable car's total horizontal travel. The calculated length of the tunnel is approximately 3784.5 meters.

Step-by-step explanation:

To find the length of the tunnel through the mountain, we can use the Pythagorean theorem. This theorem allows us to calculate the length of the hypotenuse (the tunnel in this case) of a right triangle when given the lengths of the other two sides.

The cable car's ascent and descent create a right triangle with the mountain's height as one leg and the horizontal distance between the start and end point of the cable car's journey as the other leg. The tunnel is the hypotenuse of this triangle.

Firstly, we need to find the horizontal distance:

• The cable car travels 2236 m to climb the mountain and 1414 m to descend. We can add these distances to determine the total horizontal distance the cable car travels, which is 2236 m + 1414 m = 3650 m.

Now, the Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b): c² = a² + b²

Applying this to our situation:

c² = (horizontal distance) ² + (mountain height) ²
c² = (3650 m)² + (1000 m)²
c² = 13,322,500 m² + 1,000,000 m²
c² = 14,322,500 m²
c = √14,322,500 m²
c = 3784.5 m

Therefore, the length of the tunnel is approximately 3784.5 meters.

Answer 2

3000 meters

=============

To find the length of the tunnel through the mountain, we can apply the Pythagorean theorem, since the paths the cable car takes up and down the mountain, together with the tunnel, form a right triangle.

The vertical height of the mountain is 1000 m for both the ascent and the descent, which are the heights of the two right triangles.

For the ascent, the hypotenuse is 2236 m, so we have:

• 2236² = 1000² + b²

• b² = 2236² - 1000²

• b² = 4999696 - 1000000

• b² = 3999696

• b = √3999696

• b ≈ 2000 m

For the descent, the hypotenuse is 1414 m, so we have

• 1414² = 1000² + a²

• a² = 1414² - 1000²

• a² = 1999396 - 1000000

• a² = 999396

• a = √999396

• a ≈ 1000 m

Then the length of the tunnel is the sum of 'a' and 'b', which is:

• 2000 m + 1000 m = 3000 m

Therefore, the length of the tunnel is 3000 meters.