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 45 mi hr (66 ft s). If the balloon rises vertically at a rate of 12 ft s, what is the rate of change of the distance between the motorcycle and the balloon 10 seconds later?
h = 120 + 12 t which is 240 at t = 10
dh/dt = 12
x = 66 * 10 = 660 at t = 10
dx/dt = 66
hypotenuse^2 = z^2 = h^2 + x^2
at t = 10 = 240^2 + 660^2 = 492200
so z= 702 at t = 10
2 z dz/ dt = 2 h dh/dt + 2 x dx/dt
2*702*dz/dt = 2 * 240 * 12 + 2 * 660 * 66
dz/dt = ( 2880 + 43560) / 702 = 66.2
To find the rate of change of the distance between the motorcycle and the balloon, we can use the Pythagorean theorem. Let's break it down step by step:
Step 1: Find the initial distance between the motorcycle and the balloon.
At the beginning, the motorcycle is directly beneath the balloon, so the distance between them is the height of the balloon, which is 120 ft.
Step 2: Find the rate at which the distance between the two objects is changing.
The balloon is rising vertically at a rate of 12 ft/s, which means that the balloon is moving away from the motorcycle. Therefore, the distance between them is increasing.
Step 3: Calculate the rate of change of the distance between the motorcycle and the balloon 10 seconds later.
Since the only movement is vertical, we can ignore the horizontal motion of the motorcycle. In 10 seconds, the balloon will have risen 10 * 12 = 120 ft.
Step 4: Calculate the final distance between the motorcycle and the balloon.
To determine the final distance, we need to calculate the hypotenuse of a right triangle. The initial distance between them is 120 ft, and the balloon has moved vertically 120 ft up. Therefore, the final distance is the square root of (120^2 + 120^2).
Step 5: Find the rate of change of the distance.
Now, we can find the rate of change of the distance by taking the derivative of the final distance with respect to time. Since the rate of change of the final distance is the same as the rate of change of the hypotenuse, we can apply the chain rule to find the derivative.
The derivative would be the rate of change of the final distance with respect to time, which can be calculated as follows:
d(distance)/dt = (d(sqrt(x^2 + y^2)) / dt) = (1/2(sqrt(x^2 + y^2))) * (2x(dx/dt) + 2y(dy/dt))
Where x and y are the initial distances between the motorcycle and the balloon in the x and y directions, and dx/dt and dy/dt are the rates of change of x and y, respectively.
In this case, since the motorcycle is only moving horizontally, dx/dt = 0.
d(distance)/dt = (1/2(sqrt(120^2 + 120^2))) * (2(120)(0) + 2(120)(12))
Simplifying the equation gives us:
d(distance)/dt = (1/2(sqrt(120^2 + 120^2))) * (2(120)(12))
d(distance)/dt = (1/2(sqrt(2(120^2)))) * (2(120)(12))
d(distance)/dt = (60(sqrt(2))(120)(12))
Calculating this expression will give us 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 between the motorcycle and the balloon, we need to determine the distance function relating the two objects in terms of time.
Let's start by considering the motorcycle as point A and the balloon as point B. After 10 seconds, the motorcycle would have traveled a distance of (time * velocity) = 10 * 66 = 660 feet.
At this point in time, the balloon would have risen vertically at a rate of 12 ft/s for 10 seconds, so it would be at a height of 120 + (12 * 10) = 240 feet above the ground.
Now, we have a right-angled triangle formed by the motorcycle, the balloon, and a horizontal line parallel to the ground. The distance between the motorcycle and the balloon is the hypotenuse of this triangle.
Using the Pythagorean theorem, we can find the distance between the motorcycle and the balloon:
Distance^2 = (Distance traveled by motorcycle)^2 + (Vertical distance risen by balloon)^2
Distance^2 = 660^2 + 120^2
Now, we can differentiate both sides of the equation with respect to time to find the rate of change of the distance between the motorcycle and the balloon:
2 * Distance * (Rate of change of Distance) = 2 * (660 * (Rate of change of motorcycle distance)) + 2 * (120 * (Rate of change of balloon height))
Simplifying the equation:
Distance * (Rate of change of Distance) = 660 * (Rate of change of motorcycle distance) + 120 * (Rate of change of balloon height)
We know that the rate of change of the motorcycle distance is given by the motorcycle's speed, which is 66 ft/s. The rate of change of the balloon height is given as 12 ft/s.
Substituting these values into the equation:
Distance * (Rate of change of Distance) = 660 * 66 + 120 * 12
To find the rate of change of the distance between the motorcycle and the balloon, we need to solve for (Rate of change of Distance).