At a swim meet, a parent is videotaping his son from a seat in the stands that is 20 meters past the starting line and 8 meters away from his son's lane. Let x represent the distance the son has swum.
a. Write x as a function of (theta).
b. At what angle does the parent have the camera when the race is just starting?
c. At what angle does the parent have the camera when the son has swum 25 meters.
Student note: There is a diagram with the question. I do not know if that matters too much, I do not think it will. I am very confused as to how I am supposed to set up an equation. And for part A. I just want to say that x is only part of the less than 20 meters
To solve this problem, let's start by setting up a coordinate system with the starting line as the origin (0,0) and the son's lane as the positive x-axis. The parent's seat is located 20 meters past the starting line and 8 meters away from the son's lane.
a. Write x as a function of θ:
Based on the information given, we can use trigonometry to determine x in terms of θ. Let's consider the right triangle formed by the son's position, the parent's position, and the starting line.
In this triangle, the parent's seat is the adjacent side, and the distance the son has swum (x) is the hypotenuse. The angle between the parent's seat and the son's lane (θ) can be found using trigonometric functions.
Using cosine function:
cos(θ) = adjacent/hypotenuse
cos(θ) = 8/x
Rearranging the equation, we can express x in terms of θ:
x = 8/cos(θ)
b. At what angle does the parent have the camera when the race is just starting?
When the race is just starting, the son has not swum any distance yet (x = 0). Substituting this into the equation derived in part a, we get:
0 = 8/cos(θ)
Since cosine of 0° is 1, the equation becomes:
0 = 8/1
This equation has no solution, meaning the parent cannot have the camera pointing directly at the son when the race is starting. The camera angle will be offset due to the 8 meters distance between the parent's seat and the son's lane.
c. At what angle does the parent have the camera when the son has swum 25 meters?
Substituting x = 25 into the equation derived in part a, we get:
25 = 8/cos(θ)
Rearranging the equation to solve for cos(θ), we have:
cos(θ) = 8/25
Using the inverse cosine function (arccos), we can find the value of θ:
θ = arccos(8/25)
Using a calculator, we can evaluate arccos(8/25) to find the angle θ. This will give us the camera angle at the given distance of 25 meters swum by the son.