A hot-air balloon is 120 ft above the ground when a motorcycle (traveling in a straight line on a horizontal road) passes directly beneath it going 30 mi/hr (44 ft/s). If the balloon rises vertically at a rate of 15 ft/s, what is the rate of change of the distance between the motorcycle and the balloon 6 seconds later?
it was 43.77
oops. do you see my mistake? I was considering the balloon at a constant height of 120 ft. So, in reality, at time t,
z^2 = (44t)^2 + (120+15t)^2 = 2161t^2 + 3600t + 14400
at t=6, we have z^2 = 113796
z =337.34 and
2z dz/dt = 4322t + 3600
674.68 dz/dt = 29532
dz/dt = 43.77
Well, isn't this a fun little math problem! Let's break it down, shall we?
The motorcycle is zooming beneath the hot-air balloon at 44 ft/s. That's pretty fast, considering the balloon is floating in the air. But hey, no judgments here!
Now, the balloon is not one to stay still either. It decides to rise vertically at a rate of 15 ft/s. It's like it's saying, "Hey motorcycle, I can go up too!"
So, let's find out how the distance between the motorcycle and the balloon changes after 6 seconds.
In those 6 seconds, the motorcycle travels a distance of 6 seconds * 44 ft/s = 264 ft. Talk about a speedy ride!
During the same 6 seconds, the balloon rises a distance of 6 seconds * 15 ft/s = 90 ft. Up, up, and away!
Now, let's figure out the new distance between them. Initially, the balloon was 120 ft above the ground, and the rising distance of 90 ft adds up to 120 ft + 90 ft = 210 ft.
The motorcycle, on the other hand, has traveled 264 ft horizontally. Pythagoras would be proud to help us find the new distance between them:
New distance = square root of (264^2 + 210^2)
Now we just need a calculator for that square root. Give me a moment.
Calculating...
Drumroll, please!
The new distance between the motorcycle and the balloon after 6 seconds is approximately 336.83 ft.
So, my friend, the rate of change of the distance between them 6 seconds later is 336.83 ft - (120 ft + 90 ft) = 126.83 ft.
And there you have it! The distance between the motorcycle and the balloon changed by about 126.83 ft in 6 seconds. That's faster than a clown running away from a water-spraying flower!
To find the rate of change of the distance between the motorcycle and the balloon, we need to consider the position of each object after 6 seconds.
Let's first find the final position of the motorcycle after 6 seconds:
Since the motorcycle is traveling at a constant speed of 44 ft/s, after 6 seconds, it would have traveled a distance of:
Distance = Speed * Time
Distance = 44 ft/s * 6 s = 264 ft
Now let's find the final position of the balloon after 6 seconds:
The balloon is rising vertically at a rate of 15 ft/s. Therefore, after 6 seconds, it would have risen by:
Rise = Speed * Time
Rise = 15 ft/s * 6 s = 90 ft
The initial distance between the motorcycle and the balloon is 120 ft. After 6 seconds, the positions of both the motorcycle and the balloon have changed. To find the new distance between them, we need to calculate the hypotenuse of a right triangle formed by the vertical distance (balloon's rise) and horizontal distance (distance traveled by the motorcycle).
Using the Pythagorean theorem:
New distance^2 = (Initial distance - Rise)^2 + Distance traveled^2
New distance^2 = (120 ft - 90 ft)^2 + 264 ft^2
New distance^2 = 30 ft^2 + 69,696 ft^2
New distance^2 = 69,726 ft^2
Taking the square root of both sides, we find:
New distance = √(69,726 ft^2) ≈ 264.26 ft
The rate of change of the distance between the motorcycle and the balloon after 6 seconds can be found by calculating the rate at which the new distance has changed compared to the initial distance of 120 ft:
Rate of change = (New distance - Initial distance) / Time
Rate of change = (264.26 ft - 120 ft) / 6 s
Rate of change ≈ 24.04 ft/s
Therefore, the rate of change of the distance between the motorcycle and the balloon after 6 seconds is approximately 24.04 ft/s.
as usual, draw a diagram. If the bike is x ft away, then the distance z is
z^2 = x^2 + 120^2
So that means that
z dz/dt = x dx/dt
at the moment in question, z^2 = (6*44)^2 + 120^2, so z = 290 ft
Now to find dz/dt,
290 dz/dt = (6*44) * 15
dz/dt = 396/29 ≈ 13.66 ft/s