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?
D^2 = x^2 + y^2
2 D dD/dt = 2 x dx/dt + 2 y dy/dt
or
dD/dt = (x dx/dt + y dy/dt)/D
at t = 0
x = 0
y = 150
at t = 10
x = 580
y = 150+100 = 250
D = sqrt(580^2+250^2) = 631.6
dx/dt = 58
dy/dt = 10
so plug into
dD/dt = (x dx/dt + y dy/dt)/D
Well, let's calculate it!
After 10 seconds, the motorcycle will have traveled a distance of 58 ft/sec * 10 sec = 580 ft.
During the same time, the balloon will have risen a distance of 10 ft/sec * 10 sec = 100 ft.
So, the vertical distance between the motorcycle and the balloon will remain constant at 150 ft + 100 ft = 250 ft.
Therefore, the rate of change of the distance between the motorcycle and the balloon after 10 seconds is 0 ft/sec. It looks like they'll be keeping a safe distance from each other!
And remember, always keep your eyes on the road, not on hot air balloons!
To find the rate of change of the distance between the motorcycle and the balloon 10 seconds later, we can consider the positions of the motorcycle and the balloon at that time.
Let's find the positions of both the motorcycle and the balloon after 10 seconds.
The motorcycle is traveling at a constant speed of 58 ft/sec in a straight line. Thus, after 10 seconds, the motorcycle would have traveled a distance of 10 * 58 = 580 feet.
The balloon is rising vertically at a rate of 10 ft/sec. Therefore, after 10 seconds, the balloon would have risen by 10 * 10 = 100 feet.
Now, let's calculate the final positions:
- The motorcycle has traveled 580 feet horizontally.
- The balloon has risen by 100 feet vertically.
The distance between the motorcycle and the balloon is the hypotenuse of a right triangle formed by the horizontal and vertical distances. Using the Pythagorean theorem, we can find the distance between them.
Distance^2 = Horizontal distance^2 + Vertical distance^2
Distance^2 = 580^2 + 100^2
Distance^2 = 336400 + 10000
Distance^2 = 346400
Taking the square root of both sides, we get:
Distance = sqrt(346400)
Distance ≈ 588.86 ft
Therefore, the rate of change of the distance between the motorcycle and the balloon 10 seconds later is 588.86 ft.
To find the rate of change of the distance between the motorcycle and the balloon, we need to determine how the distance is changing with respect to time. Let's break down the problem step by step:
Step 1: Determine the initial distance between the motorcycle and the balloon.
The hot air balloon is 150 ft above the ground, and when the motorcycle passes beneath it, the motorcycle is on the ground. So initially, the distance between the motorcycle and the balloon is 150 ft.
Step 2: Determine the velocity of the balloon.
The balloon is rising vertically at a rate of 10 ft/sec. So the velocity of the balloon is +10 ft/sec. A positive velocity indicates an upward direction.
Step 3: Determine the velocity of the motorcycle.
The motorcycle is traveling in a straight line on a horizontal road, so it has a horizontal velocity. The velocity of the motorcycle is given as 58 ft/sec.
Step 4: Determine the rate of change of the distance between the motorcycle and the balloon.
Let's consider the distance between the motorcycle and the balloon after t seconds. We'll call this distance D(t). We know that the balloon is rising at a rate of 10 ft/sec, so the vertical position of the balloon after t seconds will be 150 + 10t.
Now, to find the rate of change of D with respect to t, we need to find the derivative of D(t) with respect to t. The derivative of D(t) will represent the rate of change of the distance.
D(t) is the distance between the motorcycle and the balloon, which can be calculated using the Pythagorean theorem:
D(t)^2 = (horizontal distance)^2 + (vertical distance)^2
The horizontal distance is the same as the initial horizontal position of the motorcycle, which is 0 ft (since the motorcycle passed beneath the balloon). So the horizontal distance is 0.
The vertical distance is given as (150 + 10t) ft, as explained earlier.
Now, we can express D(t)^2 as:
D(t)^2 = 0^2 + (150 + 10t)^2
To find the derivative of D(t), let's differentiate both sides with respect to t:
d(D(t)^2)/dt = d(0^2 + (150 + 10t)^2)/dt
The derivative of 0 with respect to t is 0. The derivative of (150 + 10t)^2 with respect to t can be determined using the chain rule and power rule of derivatives.
Applying the chain rule and power rule, the derivative of (150 + 10t)^2 with respect to t is:
2(150 + 10t)(10)
Simplifying this expression, we get:
d(D(t)^2)/dt = 2(150 + 10t)(10)
Now, this expression represents the rate of change of D(t) squared. To find the rate of change of D(t), we need to take the square root of this expression.
Rate of change of D(t) = Sqrt(2(150 + 10t)(10))
Step 5: Determine the rate of change of the distance between the motorcycle and the balloon 10 seconds later.
To find the rate of change of the distance after 10 seconds, we substitute t = 10 into the rate of change expression.
Rate of change of D(t) at t = 10:
= Sqrt(2(150 + 10(10))(10))
= Sqrt(2(150 + 100)(10))
= Sqrt(5000)
≈ 70.71 ft/sec
Therefore, the rate of change of the distance between the motorcycle and the balloon approximately 10 seconds later is 70.71 ft/sec.