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, we can use trigonometry to find the relationship between the distance the son has swum (x) and the angles (theta) at which the parent is filming. Let's break down each part of the problem:
a. To write x as a function of (theta), we can consider the right triangle formed by the parent, the starting line, and the son. In this triangle, the side adjacent to the angle (theta) is 20 meters (the distance of the parent from the starting line) and the side opposite to the angle (theta) is x (the distance the son has swum).
By using the cosine ratio, we can write the equation:
cos(theta) = adjacent/hypotenuse
cos(theta) = 20/x
Solving for x, we get:
x = 20/cos(theta)
So, x is a function of (theta): x(theta) = 20/cos(theta)
b. In this case, when the race is just starting, the son hasn't swum any distance yet (x = 0). To find the angle (theta) at this moment, we need to rearrange the equation from part (a):
cos(theta) = 20/x
Since x = 0 at the starting line, the equation becomes:
cos(theta) = 20/0
However, dividing by zero is undefined, which means there is no angle (theta) when the race is just starting that corresponds to the position of the camera. The parent would not be able to capture any video at this exact moment.
c. To find the angle (theta) when the son has swum 25 meters, we substitute x = 25 into the equation from part (a):
cos(theta) = 20/25
cos(theta) = 4/5
Using inverse cosine (also known as arc cosine or cos^(-1)), we can find the angle (theta):
theta = cos^(-1)(4/5)
Using a calculator, the approximate value of theta is 36.87 degrees.
Therefore, when the son has swum 25 meters, the parent would have the camera positioned at an angle of approximately 36.87 degrees.
Note: While a diagram can often be helpful, the given information and equations should be sufficient to solve this problem.