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.
a. As the parent is 20 meters past the starting line and 8 meters away from his son's lane, we can use trigonometry to determine the distance the son has swum, x, as a function of (theta). Let's consider the right triangle formed by the parent, his son, and the starting line. The adjacent side of this triangle is 8 meters and the opposite side is (20 - x) meters.
Using the cosine function, the equation can be set up as:
cos(theta) = adjacent/hypotenuse
cos(theta) = 8/(20 - x)
Rearranging the equation, we get:
20 - x = 8/cos(theta)
x = 20 - 8/cos(theta)
So, x is a function of (theta) given by: x = 20 - 8/cos(theta).
b. When the race is just starting, x = 0. Plugging this value into the equation from part a, we get:
0 = 20 - 8/cos(theta)
Solving for theta, we find:
cos(theta) = 8/20
cos(theta) = 2/5
Taking the inverse cosine of 2/5, we find:
theta ≈ 66.42 degrees
Therefore, the parent has the camera at an angle of approximately 66.42 degrees when the race is just starting.
c. To determine the angle when the son has swum 25 meters, we substitute x = 25 into the equation from part a:
25 = 20 - 8/cos(theta)
Rearranging the equation, we get:
8/cos(theta) = 20 - 25
8/cos(theta) = -5
Since a cosine value cannot be negative, there is no real solution for this equation. Thus, it is not possible for the parent to have the camera at a specific angle when the son has swum 25 meters.
To solve this problem, let's consider the diagram and break down each part step-by-step:
a. Write x as a function of θ:
In the diagram, we can see that the distance between the starting line and the parent's seat is 20 meters, and the distance between the parent's seat and the son's lane is 8 meters. Let's assume the line connecting the parent's seat and the starting line is the y-axis, and the line connecting the parent's seat and the son's position is the x-axis.
From the diagram, we can see that the distance x that the son has swum is given by:
x = 20 - 8 * tan(θ)
Here, θ represents the angle at which the parent is holding the camera.
b. To find the angle θ when the race is just starting, we need to substitute x = 0 into the equation:
0 = 20 - 8 * tan(θ)
Now, solve for θ:
20 = 8 * tan(θ)
Divide both sides by 8:
2.5 = tan(θ)
To find the angle θ, take the inverse tangent (arctan) of both sides:
θ = arctan(2.5)
Use a calculator to find the value of θ. It will be approximately 68.2 degrees.
c. To find the angle θ when the son has swum 25 meters, we substitute x = 25 into the equation:
25 = 20 - 8 * tan(θ)
Now, solve for θ:
8 * tan(θ) = 20 - 25
8 * tan(θ) = -5
Divide both sides by 8:
tan(θ) = -5/8
To find the angle θ, take the inverse tangent (arctan) of both sides:
θ = arctan(-5/8)
Use a calculator to find the value of θ. It will be approximately -31.8 degrees (negative because the angle will be in the opposite direction).
To solve this question, we can use the concept of right triangles and trigonometric functions. Let's break down each part of the question:
a. Write x as a function of (theta):
In this case, we can consider a right triangle where the hypotenuse represents the distance the son has swum (x) and the opposite side represents the distance from the starting line to the parent's location (20 meters). The angle (theta) is the angle between the hypotenuse and the adjacent side.
Using trigonometry, we can use the sine function to relate the angle (theta) and the side lengths of the triangle:
sin(theta) = opposite/hypotenuse
Solving for the opposite side, we have:
opposite = sin(theta) * hypotenuse
Since the opposite side represents the distance the son has swum (x), the equation becomes:
x = sin(theta) * 20
So, x can be expressed as a function of (theta) as x = 20sin(theta).
b. At what angle does the parent have the camera when the race is just starting?
When the race is just starting, the son hasn't swum any distance yet. So, x = 0. Plugging this into the equation from part a, we have:
0 = 20sin(theta)
To solve for theta, we can use algebraic manipulation:
sin(theta) = 0
Since sin(theta) = 0 when theta = nπ (n is an integer), the angle at the start of the race can be any multiple of π.
c. At what angle does the parent have the camera when the son has swum 25 meters?
For this question, we need to find the angle (theta) when the son has swum 25 meters. Plugging x = 25 into the equation from part a:
25 = 20sin(theta)
To solve for theta, we can rearrange the equation:
sin(theta) = 25/20
Using a calculator or table of trigonometric values, we can find the inverse sine (also known as arcsine) of 25/20:
theta ≈ arcsin(25/20)
This will give us the angle in radians. To convert it to degrees, we can multiply by 180/π:
theta ≈ arcsin(25/20) * 180/π
Calculating this expression will give you the angle in degrees at which the parent has the camera when the son has swum 25 meters.