Tangent Problem

Question

A typical drag road is 402 metres long. Suppose a set of bleachers 3m deep is set up 20 m away from the track.

1. Suppose you are sitting in the front middle seat. Write an equation that represents distance, d, of the track visible by turning your head an angle of (theta) in one direction. Include the domain for this scenario.
2. Sketch the equation you determined in 1) over the domain you gave.
3. Determine the maximum and minimum values of d and θ in 1).

Note: It’s not both directions, its only one direction. Please explain everything step by step. Thank you!

Answer 1

d represents the distance along the track

tan θ = d / 20

The maximum distance along the track is 201 m

tan θ = 201 / 20

tan θ = 10.05

arctan 10.05 = θθ

θ = 84.3º

D = [0, 84.3]

R = [0, 201]

Answer 2

To determine the distance, d, of the track visible by turning your head an angle of θ in one direction, we can use trigonometry. Considering the scenario described, the bleachers are set up 20 meters away from the track, and you are sitting in the front middle seat, we can form a right triangle with the track, the bleachers, and your position. Let's assume the height of the bleachers is negligible for simplicity.

n the triangle, the hypotenuse represents the distance, d, of the track visible, and the adjacent side represents the distance from your seat to the track, which is 20 meters. The angle θ is the angle between the hypotenuse and the adjacent side.

Using the cosine function, we can write the equation as follows:

cos(θ) = Adjacent/Hypotenuse

cos(θ) = 20/d

To isolate d, we rearrange the equation:

Now let's determine the domain for this scenario. The angle θ cannot exceed the maximum angle at which the track is visible from your seat. Assuming your field of view is within the range of -90° to 90°, we can define the domain as -90° ≤ θ ≤ 90°. Note that we consider the absolute value of θ to ensure we cover both directions of turning your head.

|

d |\

| \

| \

| \

| \

| \

|______\

-90° 90°

The graph shows a curve that starts from infinity at θ = 0° and decreases as θ approaches -90° and 90°. At θ = -90° or 90°, the distance, d, becomes zero, indicating that the track is not visible in those extreme directions.

The maximum value of d occurs when θ = 0°, which is the front middle position. In this case, d = 20 meters, representing the full distance from your seat to the track.

The minimum value of d occurs at the extremes when θ = -90° or 90°, where d becomes zero

ndicating that the track is not visible in those directions.