A rotating light is located 25 feet from a wall. The light projected on the wall is moving at a rate of 1.5 feet per second when the light's angle is 15 degrees from perpendicular to the wall. If the light is turning at a constant rate, how many seconds does it take to go around once?
A = angle light is from straight at wall
x = distance along waall from that point
A = angle from that point
then
tan A = x/25
d(tan A)/dt
= sec^2 A dA/dt = (1/25)dx/dt
what we want to know is dA/dt, the turning rate
so
sec^2(15) dA/dt = (1/25)(1.5)
solve for dA/dt, the turning rate
To find out how long it takes for the light to go around once, we need to determine the circumference of the circle that the light traces.
The circumference of a circle can be calculated using the formula: C = 2πr, where C is the circumference and r is the radius of the circle.
In this case, the radius of the circle is the distance from the light to the wall, which is 25 feet.
So the circumference is C = 2π(25) = 50π feet.
To find the time it takes for the light to go around once, we need to calculate its angular speed.
The angular speed (ω) is the rate at which the light rotates, in radians per second. We know that when the light's angle is 15 degrees from perpendicular to the wall, its linear speed (v) is 1.5 feet per second.
To convert the linear speed to angular speed, we can use the formula: v = rω, where v is the linear speed, r is the radius, and ω is the angular speed.
Let's convert the linear speed from feet per second to feet per minute by multiplying by 60, since there are 60 seconds in a minute.
v = 1.5 feet per second * 60 = 90 feet per minute.
Now, we can substitute the values into the formula and solve for ω:
90 = 25 * ω
ω = 90/25 = 3.6 radians per minute.
Since we want to find the time it takes for the light to go around once, we need to convert the angular speed from radians per minute to radians per second.
There are 60 seconds in a minute, so to convert from radians per minute to radians per second, we divide by 60:
ω = 3.6 radians per minute / 60 = 0.06 radians per second.
Now, to find the time it takes for the light to go around once, we can use the formula: time = circumfe
rence/linear speed = C/v.
Plugging in the values, we get:
time = (50π) / (1.5 feet per second) = (50π) / (1.5) = 33.33 seconds (rounded to the nearest hundredth).
Therefore, it takes approximately 33.33 seconds for the light to go around once.
To solve this problem, we need to use some trigonometry and apply the concept of angular velocity.
Let's start by visualizing the situation. We have a light that is rotating around a fixed point, and the light is projected onto a wall. The distance between the light and the wall is given as 25 feet.
When the light's angle is 15 degrees from perpendicular to the wall, we can consider this as the starting point of our rotation. We want to find out how long it takes for the light to complete one full revolution.
Now, let's use trigonometry to relate the angle of rotation to the distance covered by the light on the wall.
The projection of the light on the wall creates a right-angled triangle. The distance from the light to the wall is the hypotenuse of the triangle, and the distance covered on the wall is the adjacent side. The angle formed between the hypotenuse and the adjacent side is the angle of rotation.
Given that the distance from the light to the wall is 25 feet and the angle is 15 degrees, we can use trigonometry to find the distance covered on the wall. We'll use the cosine function in this case.
cos(15) = adjacent/hypotenuse
cos(15) = distance covered on the wall / 25
Solving for the distance covered on the wall:
distance covered on the wall = 25 * cos(15)
Now, to find the time it takes for the light to complete one full rotation, we need to calculate the angular velocity.
Angular velocity is the rate at which an object rotates. It is usually measured in radians per second (rad/s). One full revolution is equal to 2π radians.
Since the distance covered on the wall is directly proportional to the angle of rotation, we can say that the distance covered on the wall is equal to the circumference of the circular path that the light follows.
Circumference = 2π * radius
The radius is equal to the distance from the light to the wall, which is 25 feet.
Circumference = 2π * 25
Now, we have the distance covered on the wall and the circumference of the circular path. We can use these two values to calculate the time it takes for the light to complete one full revolution.
Time = Distance / Speed
Since the speed is given as 1.5 feet per second, we can substitute the values into the equation:
Time = (2π * 25) / 1.5
Calculating this expression will give us the answer, which represents the number of seconds it takes for the light to go around once.