A balloon is at a height of 30 meters, and is rising at the constant rate of 5 m/sec. A bicyclist passes beneath it, traveling in a straight line at the constant speed of 10 m/sec. How fast is the distance between the bicyclist and the balloon increasing 2 seconds later?
after t seconds, the balloon is at height 30+5t
the cyclist is at a distance of 10t
so, the distance from cyclist to balloon is
z^2 = (30+5t)^2 + (10t)^2 = 125t^2 + 300t + 900
at t= 2, z = 20√5
2z dz/dt = 2(30+5t)(5) + 2(10t)(10) = 250t+300
so, at t=2,
40√5 dz/dt = 800
dz/dt = 4√5 m/s
Well, let me calculate it in a rather "amusing" way for you!
So, the balloon is rising at a rate of 5 m/sec. Now, take a moment to imagine a cyclist pedaling beneath the balloon. It's like a circus act, really.
Now, after 2 seconds, the balloon will have risen another 5m/sec × 2 sec = 10 meters. So, the height of the balloon will be 40 meters.
But what about the cyclist? Well, the cyclist will have moved 10m/sec × 2 sec = 20 meters forward.
To find the distance between them, we can use a little bit of Pythagoras. The horizontal distance between them is 20 meters, and the vertical distance is 40 meters.
Now, let's put our clownish math skills to work!
Using Pythagoras, we get:
distance^2 = horizontal distance^2 + vertical distance^2.
Plugging in our values:
distance^2 = 20^2 + 40^2.
distance^2 = 400 + 1600.
distance^2 = 2000.
And after some more math, we find that distance ≈ √2000 ≈ 44.7 meters.
Now, the question asks about how fast the distance between them is increasing.
To find that, we take the derivative of the distance formula with respect to time:
d(distance) / dt = (d(√(2000))) / dt.
Taking the derivative, we get:
d(distance) / dt = (1/2) * (2000)^(-1/2) * d(2000) / dt.
Since the rate of change of the balloon's height is given as 5 m/sec, we can say:
d(2000) / dt = 2 * 5 = 10.
Plugging this into our formula, we get:
d(distance) / dt = (1/2) * (2000)^(-1/2) * 10.
And simplifying this, we find that d(distance) / dt ≈ 0.158 m/sec.
So, after all that clownish math, we can conclude that the distance between the cyclist and the balloon is increasing at a rate of approximately 0.158 meters per second after 2 seconds. Happy cycling and balloon-watching!
To find how fast the distance between the bicyclist and the balloon is increasing 2 seconds later, we can first determine their initial distance.
At the start, the balloon is 30 meters above the ground, and the bicyclist passes beneath it. Therefore, their initial distance is 30 meters.
The balloon is rising at a constant rate of 5 m/sec, so after 2 seconds, its height will have increased by 5 m/sec × 2 sec = 10 meters. Therefore, the new height of the balloon is 30 meters + 10 meters = 40 meters.
The bicyclist is traveling at a constant speed of 10 m/sec, so after 2 seconds, the distance the bicyclist has traveled is 10 m/sec × 2 sec = 20 meters.
To calculate the distance between the balloon and the bicyclist after 2 seconds, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the two legs.
In this case, the distance between the balloon and the bicyclist forms the hypotenuse of a right triangle, with the height of the balloon representing one leg, and the distance traveled by the bicyclist representing the other leg.
Using the Pythagorean theorem, we can find the distance between the balloon and the bicyclist after 2 seconds:
Distance^2 = Height^2 + Distance Traveled^2
Distance^2 = 40^2 + 20^2
Distance^2 = 1600 + 400
Distance^2 = 2000
Taking the square root to solve for the distance:
Distance = √2000
Distance ≈ 44.72 meters
Now, to find how fast the distance between the balloon and the bicyclist is increasing 2 seconds later, we need to determine the rate at which the distance is changing with respect to time. This is the derivative of the distance with respect to time.
Differentiating both sides of the equation Distance^2 = Height^2 + Distance Traveled^2 with respect to time:
2 * Distance * d(Distance)/dt = 2 * Height * d(Height)/dt + 2 * Distance Traveled * d(Distance Traveled)/dt
Since the height of the balloon is increasing at a constant rate of 5 m/sec, d(Height)/dt = 5 m/sec.
Since the distance traveled by the bicyclist is increasing at a constant rate of 10 m/sec, d(Distance Traveled)/dt = 10 m/sec.
Plugging in the known values:
2 * 44.72 * d(Distance)/dt = 2 * 40 * 5 + 2 * 20 * 10
Simplifying the equation:
89.44 * d(Distance)/dt = 400 + 400
89.44 * d(Distance)/dt = 800
Now, divide both sides of the equation by 89.44 to solve for d(Distance)/dt:
d(Distance)/dt = 800 / 89.44
d(Distance)/dt ≈ 8.94
Therefore, the distance between the balloon and the bicyclist is increasing at a rate of approximately 8.94 meters per second, 2 seconds later.
To find how fast the distance between the bicyclist and the balloon is increasing, we can use the concept of relative motion. Let's break down the problem into its components.
The balloon is rising at a constant rate of 5 m/sec. This means that for every second that passes, the balloon moves 5 meters higher.
The bicyclist is traveling in a straight line at a constant speed of 10 m/sec. Since the balloon is directly above the bicyclist, the distance between them is essentially the vertical distance between them.
At the starting point, the balloon is 30 meters above the ground level, and the bicyclist is at the ground level.
Let's find the equation for the distance between the bicyclist and the balloon at any given time t:
d(t) = Vertical distance between them
d(t) = balloon height - bicyclist height
At time t=0, the distance between them is:
d(0) = 30 - 0 = 30 meters
Now, we want to find how fast this distance is changing after 2 seconds (t = 2).
To find the rate of change, we can take the derivative of the distance equation with respect to time:
d'(t) = d(t)/dt
Differentiating the equation d(t) = 30 - 0 = 30 with respect to t, we get:
d'(t) = 0 - 0 = 0
This means that at t = 2 seconds, the distance between the bicyclist and the balloon is not changing. It remains constant at 30 meters.