In a movie stunt, a car is driven off a cliff and falls into the ocean 200 feet below. If the car is going 50 feet/second (horizontally) when it goes over the cliff, its horizontal position is described as x = 50 t, where t is the time in seconds. Its vertical position is described as y = 16 t2, where y is the distance below the cliff (i.e. y = 200 when the car hits the ocean). Compute the length of the path traversed by the falling car. Round to the nearest foot. [Hint: The two equations will need to be combined, eliminating t so that y is expressed as a function of x.]

Question

In a movie stunt, a car is driven off a cliff and falls into the ocean 200 feet below. If the car is going 50 feet/second (horizontally) when it goes over the cliff, its horizontal position is described as x = 50 t, where t is the time in seconds. Its vertical position is described as y = 16 t2, where y is the distance below the cliff (i.e. y = 200 when the car hits the ocean). Compute the length of the path traversed by the falling car. Round to the nearest foot. [Hint: The two equations will need to be combined, eliminating t so that y is expressed as a function of x.]

Answer 1

To compute the length of the path traversed by the falling car, we need to eliminate the variable t and express the vertical position y as a function of the horizontal position x.

First, let's start with the given equations:

x = 50t
y = 16t^2

We can isolate t in the first equation by dividing both sides by 50:

t = x/50

Now we substitute this expression for t in the second equation:

y = 16(x/50)^2

Simplifying this equation gives us:

y = 0.064x^2

This equation expresses the vertical position y as a function of the horizontal position x.

Now, let's find the points where the car hits the ocean. The car hits the ocean when y equals 200, so we can set the equation equal to 200:

200 = 0.064x^2

To solve for x, we divide both sides by 0.064:

x^2 = 200 / 0.064
x^2 = 3125

Taking the square root of both sides, we get:

x = √3125
x ≈ 55.9 feet

Now we have the horizontal position x when the car hits the ocean. To find the length of the path, we need to compute the total distance traveled by the car.

The total distance traveled is the sum of the horizontal distance and vertical distance. The horizontal distance is the distance traveled horizontally, which can be calculated using the equation x = 50t. At the time of impact, t is equal to the time it takes for the car to reach the ocean. We can find this time by dividing the horizontal position x by the velocity 50 feet/second:

t = x / 50
t = 55.9 / 50
t = 1.118 seconds

The vertical distance is the vertical position y at this time:

y = 16(t^2)
y = 16(1.118^2)
y ≈ 20 feet

The total distance traveled is the sum of the horizontal and vertical distances:

Total distance = √((horizontal distance)^2 + (vertical distance)^2)
Total distance = √((55.9)^2 + (20)^2)
Total distance ≈ 59.6 feet

Therefore, the length of the path traversed by the falling car is approximately 59.6 feet, rounded to the nearest foot.

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To compute the length of the path traversed by the falling car, we need to eliminate the variable t and express the vertical position y as a function of the horizontal position x.