Find the Speed of a Car Approaching at 190mph: Optimization Problem

In summary: So when you plug in dx/dt, it's really looking at how fast the car is moving relative to the ground at that particular time.
  • #1
quatrarot
2
0
A police is flying at 150 mph at a constant altitude of .5 miles above a straight rode. The pilot uses radar to determine that an oncoming car is at a distance of exactly one mile from the helicopter, and that this distance is decreasing at 190 mph. Determine the speed of the car.

This is a problem I have tried numerous times but keep coming up with different solutions, any help would be greatly appreciated.

The Attempt at a Solution


let y = the helicopter's altitude
let x = car's distance to directly below the helicopter.
let h = the distance in miles between the helicopter and the car
let dh/dt = 190 miles/hour
150 miles/hour + dx/dt = horizontal component of car and helicopter.
then dx/dt is what we're looking for - the car's speed with respect to the ground.

y² + x² = h²

d(y²)/dt + d(x²)/dt = d(h²)/dt

dy/dt d(.5)²/dt + 150 + 2x dx/dt = d(1²)/dt 190 miles/hour = 0

2x dx/dt = 0 - 150 = -150miles/hour

x = √(1² - (.5)²) = √3/2

2(√3/2) dx/dt = -150 m/h

dx/dt = -150/√3 miles/hour

dx/dt = -86.60254 miles/hour
 
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  • #2
quatrarot said:
d(y²)/dt + d(x²)/dt = d(h²)/dt

dy/dt d(.5)²/dt + 150 + 2x dx/dt = d(1²)/dt 190 miles/hour = 0
The first equation is fine, but the second makes no sense. The 2x dx/dt term is fine... why aren't the other terms similar?
Btw, you don't actually need calculus for this problem. Just consider the components of the two speeds along the line connecting the vehicles.
 
  • #3
Okay: Then I get something like

2x*dx/dt = 190 mph

where does 150mph fit into this and is dx/dt the speed of the car?
 
  • #4
quatrarot said:
Okay: Then I get something like
2x*dx/dt = 190 mph
Not quite. You have 2y dy/dt + 2x dx/dt + 2 h dh/dt, right? So the 2's should all cancel.
Remember that x is the distance from the car to immediately below the helicopter at any given time t. This is affected by the helicopter's movement.
 

FAQ: Find the Speed of a Car Approaching at 190mph: Optimization Problem

1. What is an optimization problem?

An optimization problem is a mathematical problem in which the goal is to find the best possible solution among a set of potential solutions. In this case, the optimization problem is finding the speed of a car approaching at 190mph.

2. How can the speed of a car approaching at 190mph be found?

The speed of a car approaching at 190mph can be found by using mathematical equations and data to analyze the car's acceleration, distance, and time to determine the speed at which it is traveling.

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The accuracy of finding the speed of a car approaching at 190mph can be affected by several factors, including the precision of the data collected, the reliability of the measurement tools used, and any external factors that may affect the car's speed, such as wind or road conditions.

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Accurately determining the speed of a car approaching at 190mph is important for safety reasons. Knowing the speed at which a car is traveling can help prevent accidents and ensure that drivers are following speed limits and traffic laws.

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The results of finding the speed of a car approaching at 190mph can be applied in various real-world situations, such as designing safer roads and highways, improving car safety features, and developing more efficient transportation systems. It can also help law enforcement officers enforce speed limits and monitor traffic violations.

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