a state trooper is hidden 30ft. from a highway. One second after a truck passes, the angle beta between the highway and the line of ovservation from the patrol car to the the truck is measured.
if the speed limit is 55mph and a speeding ticket is issued for speeds of 5mph or more over the speed limit, for what angles beta should the trooper issue a ticket?
To determine for which angles beta the trooper should issue a ticket, we need to find the corresponding speeds of the passing vehicles.
Let's proceed step by step:
1. First, let's calculate the time it takes for the truck to travel a distance of 30 feet from the hidden state trooper. We know that the speed limit is 55 mph.
The time t it takes for the truck to travel 30 ft can be calculated using the formula:
Distance = Speed x Time
Rearranging the formula to solve for time:
Time = Distance / Speed
Converting the speed limit from mph to ft/s:
Speed = 55 mph * 1.46667 ft/s/mph = 80.67 ft/s
Substituting the values into the formula:
Time = 30 ft / 80.67 ft/s ≈ 0.37 seconds
Therefore, it takes the truck approximately 0.37 seconds to travel the 30-feet distance from the hidden state trooper.
2. Now that we have the time it takes for the truck to pass the hidden state trooper, we need to determine the trigonometric relationship between the angle beta and the speed of the truck.
Let's imagine a right triangle where the base represents the distance of 30 ft between the trooper and the highway, and the line of observation from the trooper to the truck represents the hypotenuse. The angle beta is the angle between the highway and the line of observation.
Using trigonometry, we can say that:
tan(beta) = Opposite / Adjacent
Since we want to find the relationship between beta and the speed of the truck, we can substitute the opposite side (30 ft) with the speed multiplied by time, as the truck's speed multiplied by the time gives the distance traveled during that time.
Thus:
tan(beta) = (Speed of the truck) x (Time)
3. To find the speeds of the vehicles for which the trooper should issue a ticket, we need to calculate the tangent of the angles beta and compare it to the threshold value for issuing a ticket.
The trooper issues a ticket for speeds of 5 mph or more over the speed limit, which is 55 mph. Therefore, the threshold speed is 60 mph.
If we rearrange the angle beta formula from step 2, we can solve for the speed of the truck:
Speed of the truck = tan(beta) / (Time)
Substituting the values:
Speed of the truck = tan(beta) / 0.37
We can iterate through different beta angles, calibrate the tangent value, and compare it to the threshold speed of 60 mph. For any beta angle that yields a speed greater than 60 mph, the trooper should issue a ticket.
So, for what angles beta should the trooper issue a ticket? For any angle beta that satisfies the following condition:
tan(beta) / 0.37 > 60 mph
To find the specific angles beta, you would need to solve the inequality above by isolating beta and find the corresponding values satisfying the condition.
Note: It's important to note that the above calculation assumes no deceleration or acceleration of the truck.