A roller coaster has a track that drops at an angle of depression of 25 degrees from a height of 14.9 m. When it reaches the ground in travels horizontally for 8 m. It then rises at an angle of elevation of 47 degrees to a height of 26.8 m

A) What is the total horizontal distance covered by this portion of track?
B) What is the total distance travelled by a car on this portion of the roller coaster track?

A) The total horizontal distance covered by this portion of the track is the sum of the horizontal distance when it drops and when it travels horizontally.

Let's calculate:

Using trigonometric functions, we can find the horizontal distance when it drops:
h = 14.9 m
angle of depression = 25 degrees

The horizontal distance is given by:
horizontal distance = h * tan(angle of depression)

horizontal distance = 14.9 * tan(25) ≈ 6.581 m

When it travels horizontally for 8 m, we can directly add this distance to the previous horizontal distance:

total horizontal distance = 6.581 + 8 = 14.581 m

Therefore, the total horizontal distance covered by this portion of the track is approximately 14.581 m.

B) The total distance traveled by the car on this portion of the roller coaster track is the sum of the vertical distance and the horizontal distance.

The vertical distance is the rise of the roller coaster when it reaches a height of 26.8 m, given an angle of elevation of 47 degrees.

Using trigonometric functions, we can find the vertical distance:
angle of elevation = 47 degrees
vertical distance = h / tan(angle of elevation)

vertical distance = 26.8 / tan(47) ≈ 26.8 / 1.0724 ≈ 25 m

Therefore, the vertical distance covered by this portion of the track is approximately 25 m.

The total distance traveled is the sum of the horizontal and vertical distances:
total distance = horizontal distance + vertical distance

total distance = 14.581 + 25 ≈ 39.581 m

Therefore, the total distance traveled by the car on this portion of the roller coaster track is approximately 39.581 m.

To determine the total horizontal distance covered by this portion of the track, we will use trigonometry.

A) Total horizontal distance:

First, we need to find the vertical distance covered by the roller coaster in the first part of the track.

Vertical distance = initial height - final height
= 14.9 m - 0 m (as it reaches the ground)

Vertical distance = 14.9 m

Now, we can find the total horizontal distance using the angle of depression and the vertical distance.

Horizontal distance = vertical distance * tan(angle of depression)

Horizontal distance = 14.9 m * tan(25°)

Horizontal distance ≈ 6.729 m

Therefore, the total horizontal distance covered by this portion of the track is approximately 6.729 meters.

B) Total distance traveled:

To find the total distance traveled by the car on this portion of the roller coaster track, we need to find the hypotenuse of the first part of the track and the second part of the track.

For the first part of the track:

Hypotenuse = square root of (horizontal distance^2 + vertical distance^2)

Hypotenuse = √(6.729^2 + 14.9^2)

Hypotenuse ≈ 16.65 meters

For the second part of the track:

Hypotenuse = vertical distance / sin(angle of elevation)

Hypotenuse = 26.8 m / sin(47°)

Hypotenuse ≈ 38.43 meters

Therefore, the total distance traveled by the car on this portion of the roller coaster track is approximately 16.65 meters (from the first part of the track) + 38.43 meters (from the second part of the track) ≈ 55.08 meters.

To solve this problem, we can use trigonometry and geometry concepts. Let's break down the problem into smaller parts and solve it step by step.

A) To find the total horizontal distance covered by this portion of the track, we need to determine the horizontal distance covered during the drop and the horizontal distance covered during the rise.

1. The angle of depression during the drop is 25 degrees, and the height dropped is 14.9 m. The horizontal distance covered during the drop can be found using the trigonometric function tangent (tan).

tan(25 degrees) = Opposite / Adjacent
tan(25 degrees) = horizontal distance / 14.9 m

To solve for the horizontal distance, multiply both sides by 14.9:

horizontal distance = tan(25 degrees) * 14.9 m

Using a calculator, we can compute this value:

horizontal distance = 6.54 m (approximately)

2. The roller coaster then travels horizontally for 8 m. This horizontal distance is already given.

3. Finally, the roller coaster rises at an angle of elevation of 47 degrees to a height of 26.8 m. Similar to the drop, we can calculate the horizontal distance covered during the rise using the tangent function.

tan(47 degrees) = horizontal distance / 26.8 m

Solving for the horizontal distance:

horizontal distance = tan(47 degrees) * 26.8 m

Using a calculator, we can compute this value:

horizontal distance = 23.24 m (approximately)

Now, to find the total horizontal distance covered, we add the horizontal distances of the drop, the horizontal section, and the rise:

Total horizontal distance covered = 6.54 m + 8 m + 23.24 m

Total horizontal distance covered = 37.78 m

Therefore, the total horizontal distance covered by this portion of the track is 37.78 meters.

B) To find the total distance traveled by a car on this portion of the roller coaster track, we need to consider the vertical distances as well.

1. During the drop, the vertical distance is 14.9 m.

2. During the horizontal section, there is no change in vertical distance, so the vertical distance covered is 0 m.

3. During the rise, the vertical distance is 26.8 m.

Now, to find the total distance traveled, we use the Pythagorean theorem, which states that the square of the hypotenuse (the total distance traveled) is equal to the sum of the squares of the other two sides (vertical and horizontal distances).

Using the values we calculated earlier:

Total distance traveled = √((14.9 m)^2 + (37.78 m)^2 + (26.8 m)^2)

Using a calculator, we can compute this value:

Total distance traveled = 48.09 m (approximately)

Therefore, the total distance traveled by a car on this portion of the roller coaster track is 48.09 meters.

(A) 19 cot25° + 26.8 cot47°

(B) 19 csc25° + 26.8 csc47°

Draw a diagram and review your basic trig function definitions