Question

the angle of elevation from the bottom of a scenic gondola ride to the top of a mountain is 31 degrees. if the vertical distance from the bottom to the top of the mountain is 902 feet and the gondola moves at a speed of 155 feet per minute, how long does the ride last? round to the nearest minute

Answer 1

The gondola ride lasts approximately 11 minutes.

Step-by-step explanation:

To find the duration of the gondola ride, we need to calculate the time it takes for the gondola to reach the top of the mountain. We can use the angle of elevation and the vertical distance to determine the horizontal distance traveled by the gondola. Then, we can divide the horizontal distance by the speed of the gondola to find the time.

Using trigonometry, we can find that the horizontal distance is given by: horizontal distance = vertical distance / tan(angle of elevation).

Substituting the given values: horizontal distance = 902 feet / tan(31 degrees) = 1654.88 feet.

Now, we can divide the horizontal distance by the speed of the gondola: time = horizontal distance / speed = 1654.88 feet / 155 feet per minute ≈ 10.67 minutes.

Rounding to the nearest minute, the ride lasts approximately 11 minutes.

Answer 2

The ride lasts 10 minutes.

Step-by-step explanation:

The triangle that is formed is attached.

In order to find out how long the ride lasts, we need to figure out the horizontal distance
`d.`

From trigonometry we have:

`tan(31^o)=(902)/(d).`

Therefore

`d=(902\:feet)/(tan(31^o))=1501\:feet.`

Now the amount of time
`t` the gondola ride lasts is equal to the distance
`d` divided by the speed of the gondola:

`t=(1501ft)/(155ft/sec) =\boxed{9.69\:minutes.}`

To the nearest minute this is 10 minutes.