A baloon is 200 m off the ground and rising vertically at the rate of 15 m/sec an automobile passes beneath it travelling along a straight road at a constant rate of 45km/hour how fast is the distance between them changing one second later???

Question

A baloon is 200 m off the ground and rising vertically at the rate of 15 m/sec an automobile passes beneath it travelling along a straight road at a constant rate of 45km/hour how fast is the distance between them changing one second later???

Answer 1

after t seconds, the distance d is

d^2 = (200+15t)^2 + (12.5t)^2
d^2 = 381.25t^2 + 960t + 6400
So,

2d dd/dt = 762.5t + 960
At t=1, d=215.36

So, at t=1, dd/dt = 4.0 m/s

Answer 2

No, that can't be right. The balloon is rising at 15 m/s.

Aha.

d^2 = 381.25t^2 + 6000t + 40000
2d dd/dt = 762.5t + 6000
dd/dt = 15.7 m/s

Answer 3

To find how fast the distance between the balloon and the automobile is changing, we need to determine the rate at which the automobile is moving towards the balloon.

Given:
- The balloon is rising vertically at a rate of 15 m/sec.
- The automobile is traveling at a constant rate of 45 km/hour.

First, we need to convert the automobile's speed from km/h to m/sec:
1 km = 1000 m
1 hour = 3600 seconds

So, the automobile's speed in m/sec is:
45 km/hour * (1000 m / 1 km) * (1 hour / 3600 seconds) = 12.5 m/sec

Next, let's calculate the distance between the balloon and the automobile after one second. We can use the formula:

Distance = Initial Distance + (Speed × Time)

The initial distance between the balloon and the automobile is 200 meters. The automobile's speed towards the balloon is 12.5 m/sec. The time is 1 second.

Distance = 200 m + (12.5 m/sec × 1 sec) = 200 m + 12.5 m = 212.5 m

Therefore, the distance between the balloon and the automobile one second later will be 212.5 meters.

Answer 4

To find how fast the distance between the balloon and the automobile is changing one second later, we need to use related rates.

Let's assume the distance between the balloon and the automobile is represented by the variable "d." We know that the balloon is rising vertically at a rate of 15 m/sec, and the automobile is traveling along a straight road at a constant rate of 45 km/hour.

First, we need to convert the automobile's speed to the same unit as the balloon's speed, which is meters per second.

45 km/hour = 45,000 meters / 3,600 seconds = 12.5 meters/second.

Now, we can establish the relationship between the variables:

d^2 = (horizontal distance)^2 + (vertical distance)^2

The horizontal distance remains constant and equal to zero because the balloon and the automobile are in a vertical line. So, we only need to consider the vertical distance.

Using the Pythagorean theorem, we have:

d^2 = 0 + (200 + 15t)^2

Expanding and differentiating both sides with respect to time gives us:

2d * (dd/dt) = 2(200 + 15t)(15)

Simplifying the equation, we have:

dd/dt = (200 + 15t)(15) / d

To calculate the speed of the distance changing one second later, substitute the current time (t = 1 second), and the initial distance (d = 200 m) into the equation:

dd/dt = (200 + 15 * 1)(15) / 200

dd/dt = (200 + 15)(15) / 200

dd/dt = (215)(15) / 200

dd/dt = 1612.5 / 200

dd/dt ≈ 8.06 m/sec

Therefore, the distance between the balloon and the automobile is changing at a rate of approximately 8.06 meters per second one second later.