A hot air balloon is 150 ft above the ground when a motorcycle passes beneath it (traveling in a striaght line on a horizontal road) going 58 ft/sec. If the balloon is rising vertically at a rate of 10 ft/sec, what is the rate of change of the distance between the motorcycle and the balloon 10 seconds later?
Let t sec be the time since the balloon was directly above the bike, and let d ft be the distance between the balloon and the bike.
According to my diagram,
d^2 = (58t)^2 + (150 + 10t)^2
= 3364t^2 + 22500 + 3000t + 100t^2
= 3464t^2 + 3000t
2d dd/dt = 6928t + 3000
dd/dt = (3464t + 1500)/d
plug t = 10 into the d^2 = ... equation, and find d
now plut t = 10, and d = ....
into the dd/dt expression , remember the units would be ft/s
To find the rate of change of the distance between the motorcycle and the balloon after 10 seconds, we need to consider the velocities of both the balloon and the motorcycle.
Let's first calculate the distance between the balloon and the motorcycle initially. Since the motorcycle is passing beneath the balloon, the distance between them is the vertical distance from the balloon to the ground, which is 150 ft.
After 10 seconds, the balloon would have risen vertically at a rate of 10 ft/sec for 10 seconds, so the balloon would be at a new height of:
150 ft + (10 ft/sec * 10 sec) = 150 ft + 100 ft = 250 ft.
During this 10-second interval, the motorcycle is moving horizontally at a constant rate of 58 ft/sec, so the motorcycle would have traveled a horizontal distance of:
58 ft/sec * 10 sec = 580 ft.
Now, we have a right triangle formed by the vertical distance between the motorcycle and the balloon (250 ft) and the horizontal distance traveled by the motorcycle (580 ft). We can use the Pythagorean theorem to find the distance between them.
Distance^2 = Vertical distance^2 + Horizontal distance^2
Distance^2 = 250 ft^2 + 580 ft^2
Distance^2 = 62500 ft^2 + 336400 ft^2
Distance^2 = 398900 ft^2
Taking the square root of both sides, we find:
Distance = √398900 ft
Distance ≈ 631.39 ft
Therefore, the distance between the motorcycle and the balloon after 10 seconds is approximately 631.39 ft.
Finally, to find the rate of change of this distance, we differentiate the distance equation with respect to time (t):
d(distance)/dt = (d/dt)√(250^2 + (58*t)^2)
Using the chain rule, we get:
d(distance)/dt = (0 + 58^2*t)/(2 * √(250^2 + (58*t)^2))
So, the rate of change of the distance between the motorcycle and the balloon after 10 seconds can be found by substituting t = 10 into the derivative equation above:
d(distance)/dt ≈ (58^2 * 10)/(2 * √(250^2 + (58 * 10)^2))
To find the rate of change of the distance between the motorcycle and the balloon 10 seconds later, we need to calculate the positions of the motorcycle and the balloon after 10 seconds.
Let's first calculate the position of the motorcycle after 10 seconds:
Since the motorcycle is travelling at a constant speed of 58 ft/sec, after 10 seconds it would have traveled a distance of 58 ft/sec * 10 sec = 580 ft.
Next, let's calculate the position of the balloon after 10 seconds:
The balloon's vertical rise rate is 10 ft/sec, so after 10 seconds, the balloon would have risen by 10 ft/sec * 10 sec = 100 ft.
The initial height of the balloon was 150 ft, so after 10 seconds, the balloon would be at a height of 150 ft + 100 ft = 250 ft.
Now, we can find the rate of change of the distance between the motorcycle and the balloon after 10 seconds.
The distance between the motorcycle and the balloon is the difference between their vertical positions.
The vertical position of the motorcycle after 10 seconds is 0 ft since it is on the ground level.
The vertical position of the balloon after 10 seconds is 250 ft.
Therefore, the rate of change of the distance between the motorcycle and the balloon after 10 seconds is the difference between their vertical positions over time, which is 250 ft - 0 ft = 250 ft.