Derivative as a rate of change exercise

[thegreengineer]

Homework Statement

A police car is parked 50 feet away from a wall. The police car siren spins at 30 revolutions per minute. What is the velocity the light moves through the wall when the beam forms angles of: a) α= 30°, b) α=60°, and c) α=70°?

This is the diagram:
https://scontent-a-lax.xx.fbcdn.net/hphotos-xpf1/v/t1.0-9/10665816_1485403331744206_1785843909705272494_n.jpg?oh=8c58e875096c8045213bef3bff034736&oe=54BD392F
Where:

• α is the angle the light beam makes with the siren (and it is a variable)
• dα/dt is the angular velocity of the light beam (and it is a constant; 30 revolutions per minute)
• x is the length of what the beam hits in the wall (it is a variable)
• dx/dt is velocity on how much light is being struck in the wall (it is a variable, and what's we are supposed to compute with the given data)
• Also we have the distance between the police car and the wall, which is also a constant and is 50 feet.

Ok, before going to the formulas, according to my textbook the answers should be:

• When the angle is 30° (= 0.524 rad) the velocity is 200π/3 feet per second
• When the angle is 60° (= 1.047 rad) the velocity is 200π feet per second
• When the angle is 70° (= 1.222 rad) the velocity is approximately 427.43π feet per second

Homework Equations

Well, this is a calculus problem, it does not take a pre-shaped formula as in physics or chemistry. However, we are dealing with derivatives as a rate of change. I found the relationship which may lead to the problem answer:

If we consider the previous diagram the formulas are:

• x=50tan(α) which leads to the one below
• dx/dt=50(sec²α)(dα/dt)= 50(1/cos²α) (this is the equation I used for attempting to solve the problem)

NOTE: For getting the second equation, I used the trigonometric derivatives rules

The Attempt at a Solution

I did use the last formula; replacing with the given data we get:

For the angle of ,524 rad dx/dt=50(1/cos²(,524))=66.69 ft/s which is not the same as 200π/3 ft/s
For the angle of 1,047 rad dx/dt=50(1/cos²(,524))=201.96 ft/s which is not the same as 200π ft/s
For the angle of 1,222 rad dx/dt=50(1/cos²(,524))=423.4 ft/s which is not the same as 427.43 ft/s

I need to see what I'm doing wrong; I've been stuck in this problem for about an hour and a half :(

 

[Mark44]

Homework Statement

A police car is parked 50 feet away from a wall. The police car siren spins at 30 revolutions per minute. What is the velocity the light moves through the wall when the beam forms angles of: a) α= 30°, b) α=60°, and c) α=70°?

This is the diagram:
https://scontent-a-lax.xx.fbcdn.net/hphotos-xpf1/v/t1.0-9/10665816_1485403331744206_1785843909705272494_n.jpg?oh=8c58e875096c8045213bef3bff034736&oe=54BD392F
Where:

• α is the angle the light beam makes with the siren (and it is a variable)
• dα/dt is the angular velocity of the light beam (and it is a constant; 30 revolutions per minute)
• x is the length of what the beam hits in the wall (it is a variable)
• dx/dt is velocity on how much light is being struck in the wall (it is a variable, and what's we are supposed to compute with the given data)
• Also we have the distance between the police car and the wall, which is also a constant and is 50 feet.

Ok, before going to the formulas, according to my textbook the answers should be:

• When the angle is 30° (= 0.524 rad) the velocity is 200π/3 feet per second
• When the angle is 60° (= 1.047 rad) the velocity is 200π feet per second
• When the angle is 70° (= 1.222 rad) the velocity is approximately 427.43π feet per second

Homework Equations

Well, this is a calculus problem, it does not take a pre-shaped formula as in physics or chemistry. However, we are dealing with derivatives as a rate of change. I found the relationship which may lead to the problem answer:

If we consider the previous diagram the formulas are:

• x=50tan(α) which leads to the one below
• dx/dt=50(sec²α)(dα/dt)= 50(1/cos²α) (this is the equation I used for attempting to solve the problem)

In your formula for dx/dt, it looks like you have lost dα/dt at the end.

NOTE: For getting the second equation, I used the trigonometric derivatives rules

The Attempt at a Solution

I did use the last formula; replacing with the given data we get:

For the angle of ,524 rad dx/dt=50(1/cos²(,524))=66.69 ft/s which is not the same as 200π/3 ft/s
For the angle of 1,047 rad dx/dt=50(1/cos²(,524))=201.96 ft/s which is not the same as 200π ft/s
For the angle of 1,222 rad dx/dt=50(1/cos²(,524))=423.4 ft/s which is not the same as 427.43 ft/s

I need to see what I'm doing wrong; I've been stuck in this problem for about an hour and a half :(

 

[LCKurtz]

In addition to Mark44's comments, don't change everything to decimals. Use $30^\circ = \frac \pi 6$radians and $30$rpm$=\pi$rad/sec.

 

[thegreengineer]

Thanks everyone, I checked and I got it right.