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
Wow connor do it ur self
a) Tata = tan^-1(20/8) - tan^-1((20-X)/8)
To solve this problem, let's break it down step by step:
a. Write x as a function of θ:
From the diagram, we can see that the distance the son has swum (x) forms one side of a right triangle. The parent's seat in the stands, 20 meters past the starting line, forms the hypotenuse of the triangle, and the distance from the parent's seat to the son's lane, 8 meters, forms the adjacent side.
Using trigonometry, we can relate these sides using the cosine function:
cos(θ) = (8 meters) / (20 meters)
cos(θ) = 0.4
To isolate θ, we take the inverse cosine (arccos) of 0.4:
θ = arccos(0.4)
Thus, x can be written as a function of θ: x = 20 meters * 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, so x = 0. Plugging this into the equation from part a, we can solve for θ:
0 = 20 meters * cos(θ)
cos(θ) = 0
Since cos(θ) = 0 when θ = 90 degrees or π/2 radians, the parent would have the camera at a 90-degree angle when the race starts.
c. At what angle does the parent have the camera when the son has swum 25 meters:
Now, we need to find the angle at which the parent has the camera when the son has swum 25 meters. We can again use the equation from part a:
x = 20 meters * cos(θ)
Plugging in x = 25 meters, we can solve for θ:
25 meters = 20 meters * cos(θ)
cos(θ) = 25 meters / 20 meters
cos(θ) = 1.25
Taking the inverse cosine of 1.25, we find:
θ ≈ arccos(1.25)
Since the cosine function is only defined for values between -1 and 1, there is no real solution for θ in this case. Therefore, the parent cannot have the camera at a valid angle when the son has swum 25 meters.
Keep in mind that if there are any further details or specific instructions in the diagram, it's always best to consider them to ensure an accurate answer.