A hot-air balloon is 180 ft above the ground when a motorcycle passes directly beneath it (travelling in a straight line on a horizontal road) going at a constant speed of 60 ft/s. If the balloon is rising vertically at a rate of 15 ft/s, what is the rate of change of the distance between the motorcycle and the balloon 20 seconds after the motorcycle was directly beneath the balloon.
Well, this is quite a situation! It's like a circus act in the making - a motorcycle and a balloon playing a game of chase in the sky! Let's see if we can unravel this entertaining math puzzle.
We have a hot-air balloon rising at a rate of 15 ft/s and a motorcycle zooming along a horizontal road at a constant speed of 60 ft/s. Now, we need to find the rate of change of the distance between them 20 seconds after the motorcycle was directly beneath the balloon.
Let's break it down step by step. Initially, when the motorcycle passes beneath the balloon, they are at the same point vertically. The distance between them is 180 ft, but that won't stay constant since the balloon is also rising.
Since the balloon is going up at a rate of 15 ft/s, after 20 seconds, it will have risen an additional distance of 15 ft/s * 20 s = 300 ft. So, after 20 seconds, the height of the balloon will be 180 ft + 300 ft = 480 ft.
Now, let's consider the horizontal distance between the motorcycle and the balloon. The motorcycle is moving at a constant speed of 60 ft/s, so after 20 seconds, it will have traveled a distance of 60 ft/s * 20 s = 1200 ft.
Therefore, the distance between the balloon and the motorcycle after 20 seconds will be the hypotenuse of a right triangle with one side measuring 480 ft (the height of the balloon) and the other side measuring 1200 ft (the horizontal distance traveled by the motorcycle).
To find the rate of change of this distance, we can use the Pythagorean theorem:
distance^2 = (height)^2 + (horizontal distance)^2
Differentiating both sides with respect to time:
2 * distance * (rate of change of distance) = 2 * height * (rate of change of height) + 2 * horizontal distance * (rate of change of horizontal distance)
Since both the height and horizontal distance are constant (the balloon is rising vertically and the motorcycle is moving on a straight line), their rate of changes are zero.
Simplifying the equation, we find:
(rate of change of distance) = 0
So, after 20 seconds, the rate of change of the distance between the motorcycle and the balloon is zero. It means that the distance between them will remain constant.
Looks like the motorcycle won't be able to catch up with the balloon in this circus act!
To find the rate of change of the distance between the motorcycle and the balloon, we need to consider the vertical and horizontal components separately.
Let's first consider the horizontal component. Since the motorcycle is traveling in a straight line on a horizontal road, its horizontal distance from the balloon remains constant. Therefore, the horizontal component does not affect the rate of change of the distance between the two.
Next, let's consider the vertical component. The balloon is rising vertically at a rate of 15 ft/s, so its height increases by 15 ft every second.
We are given that the motorcycle passes directly beneath the balloon when it is 180 ft above the ground. After t seconds, the balloon's height from the ground is given by h(t) = 180 + 15t.
To find the rate of change of the distance between the motorcycle and the balloon, we can calculate the derivative of the distance between the two with respect to time.
Let d(t) be the distance between the motorcycle and the balloon at time t. We can find d(t) using the Pythagorean theorem:
d(t) = √[h(t)^2 + x^2]
where x is the horizontal distance between the motorcycle and the balloon (which remains constant).
Taking the derivative of d(t) with respect to time, we get:
d'(t) = (h'(t) * h(t)) / √[h(t)^2 + x^2]
= (15 * (180 + 15t)) / √[(180 + 15t)^2 + x^2]
Now, we need to find the value of x, the horizontal distance between the motorcycle and the balloon. The motorcycle is traveling at a constant speed of 60 ft/s, and the time it takes to travel from directly beneath the balloon to a point 20 seconds later is 20 seconds. Therefore, the horizontal distance covered by the motorcycle is:
x = 60 * 20 = 1200 ft
Substituting this value into our equation for d'(t), we get:
d'(t) = (15 * (180 + 15t)) / √[(180 + 15t)^2 + 1200^2]
Now, we can calculate the rate of change of the distance between the motorcycle and the balloon 20 seconds after the motorcycle was directly beneath the balloon by substituting t = 20 into the equation:
d'(20) = (15 * (180 + 15 * 20)) / √[(180 + 15 * 20)^2 + 1200^2]
Simplifying this expression will give us the rate of change of the distance between the motorcycle and the balloon 20 seconds later.
To find the rate of change of the distance between the motorcycle and the balloon, we can use the Pythagorean theorem. Let's break down the problem step by step:
Step 1: Find the initial horizontal distance between the motorcycle and the balloon.
Since the motorcycle passes directly beneath the balloon, the initial horizontal distance is zero.
Step 2: Determine the vertical distance between the motorcycle and the balloon.
Given that the balloon is rising vertically at a rate of 15 ft/s, the vertical distance will increase by 15 ft for every second that passes.
Step 3: Calculate the distance between the motorcycle and the balloon using the Pythagorean theorem.
The distance between the motorcycle and the balloon is the hypotenuse of a right triangle formed by the horizontal and vertical distances. Let's denote this distance as D.
We can write the equation:
D^2 = horizontal distance^2 + vertical distance^2
Initially, the horizontal distance is zero, so the equation becomes:
D^2 = 0^2 + (180 ft + 15 ft/s * t)^2
Step 4: Differentiate both sides of the equation to find the rate of change of the distance between the motorcycle and the balloon.
Differentiating the equation with respect to time t gives us:
2D * dD/dt = 0 + 2 * (180 ft + 15 ft/s * t) * (15 ft/s)
Simplifying the equation, we get:
2D * dD/dt = 2 * (180 ft + 15 ft/s * t) * (15 ft/s)
Step 5: Substitute t = 20 seconds into the equation and solve for dD/dt.
Substituting t = 20 into the equation, we have:
2D * dD/dt = 2 * (180 ft + 15 ft/s * 20 s) * (15 ft/s)
= 2 * (180 ft + 300 ft) * (15 ft/s)
= 2 * 480 ft * 15 ft/s
= 14,400 ft^2/s
Now, we can solve for dD/dt:
dD/dt = 14,400 ft^2/s / (2D)
= 7,200 ft^2/sD
So, the rate of change of the distance between the motorcycle and the balloon 20 seconds after the motorcycle was directly beneath the balloon is 7,200 ft^2/sD.
Wow, this is such an old question. The answer is easy, namely:
during the 20 seconds that passed:
- baloon increased altitude by 300ft,
- car drove 1200ft and so:
Initial distance between them was 180ft and after 20 seconds
it s squareRoot(power(480,2) + power(1200,2)), which is 1292,44ft.
Distance changed by 1112,44ft
draw the figure, a right triangle. figure out distances after 20seconds.
let horizontal distance be x, height be h, and d the slant distance.
d^2=h^2+x^2
2d dd/dt= 2h dh/dt+2x dx/dt
solve for dd/dt, you know all else.