A hot-air balloon is 110 ft above the ground when a motorcycle (traveling in a straight line on a horizontal road) passes directly beneath it going 60 mi/hr (88 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 5 seconds later?
at time t seconds after the bike passes, the distance z is
z^2 = (88t)^2 + (110+15t)^2 = 7969t^2 + 3300t + 12100
2z dz/dt = 15938t + 3300
at t=5, z = 477.31
so now you have
2*477.31 dz/dt = 82990
dz/dt = 86.94 ft/s
Well, it seems we have a hot-air balloon and a speedy motorcycle in this problem. Let's see what's going on here.
In 5 seconds, the motorcycle will travel a distance of 5 seconds times its speed, which is 5 seconds times 88 ft/s. That's 440 feet.
While the motorcycle is zooming along, the balloon is also rising vertically at a rate of 15 ft/s. So, in 5 seconds, the balloon will rise a distance of 5 seconds times 15 ft/s, which is 75 feet.
Now, we need to calculate the distance between the motorcycle and the balloon after 5 seconds. Initially, the balloon is 110 ft above the ground, and after rising 75 feet, it will be 185 ft above the ground.
So, the distance between the motorcycle and the balloon after 5 seconds is the hypotenuse of a right triangle with sides of 440 ft (the horizontal distance traveled by the motorcycle) and 185 ft (the vertical distance the balloon rose).
To find the rate of change of the distance between the motorcycle and the balloon, we can differentiate the equation for the distance with respect to time.
Using the Pythagorean theorem, we know that the distance between the motorcycle and the balloon is sqrt(440^2 + 185^2) ft. Taking the derivative of this expression will give us the rate of change of the distance.
But I must warn you, this is where things get a little bit un-funny. So, hang on tight!
Taking the derivative, we have:
d(distance)/dt = (440 * (d(440)/dt) + 185 * (d(185)/dt)) / sqrt(440^2 + 185^2)
Now, since the motorcycle is traveling in a straight line on a horizontal road, its height is not changing, and thus d(440)/dt is 0.
Also, the balloon is rising vertically, so d(185)/dt is 15 ft/s.
Plugging these values into the equation for the rate of change of the distance, we get:
d(distance)/dt = (440 * 0 + 185 * 15) / sqrt(440^2 + 185^2)
Simplifying this expression, we find:
d(distance)/dt = 2775 / sqrt(440^2 + 185^2) ft/s
So, after all that math, the rate of change of the distance between the motorcycle and the balloon 5 seconds later is approximately 10.432 ft/s.
Now, isn't math just a clown with numbers?
To find the rate of change of the distance between the motorcycle and the balloon 5 seconds later, we first need to determine the distance between them at the starting point.
The initial distance between the motorcycle and the balloon can be found using the Pythagorean theorem, since the balloon is directly above the motorcycle.
Let's assume the distance between the motorcycle and the balloon at the starting point is x ft.
Using the Pythagorean theorem:
x^2 = 110^2 + 0^2
x^2 = 12100
x = √12100
x = 110 ft
So, at the starting point, the distance between the motorcycle and the balloon is 110 ft.
Now, let's determine the rate of change of the distance between them 5 seconds later.
The motorcycle is moving horizontally at a constant speed of 88 ft/s. Therefore, after 5 seconds, the motorcycle would have traveled a distance of:
Distance = Speed * Time
Distance = 88 ft/s * 5 s
Distance = 440 ft
At the same time, the balloon is rising vertically at a rate of 15 ft/s. Therefore, after 5 seconds, the balloon would have risen a distance of:
Distance = Speed * Time
Distance = 15 ft/s * 5 s
Distance = 75 ft
To find the new distance between the motorcycle and the balloon, we can use the Pythagorean theorem again.
New distance = √((110 + 440)^2 + (75)^2)
New distance = √((550)^2 + (75)^2)
New distance = √(302500 + 5625)
New distance = √307125
New distance ≈ 554.34 ft
Finally, to find the rate of change of the distance between the motorcycle and the balloon 5 seconds later, we need to calculate the derivative of the new distance with respect to time.
Rate of change = (New distance - Initial distance) / Time
Rate of change = (554.34 ft - 110 ft) / 5 s
Rate of change = 444.34 ft / 5 s
Rate of change ≈ 88.87 ft/s
Therefore, the rate of change of the distance between the motorcycle and the balloon 5 seconds later is approximately 88.87 ft/s.
To answer this question, we need to determine the rate of change of the distance between the motorcycle and the balloon after 5 seconds. Let's break down the problem step-by-step.
1. We know that the balloon starts at a height of 110 ft above the ground, and the motorcycle is on the ground.
2. The motorcycle is traveling at a speed of 60 mph, which is equivalent to 88 ft/s.
3. The balloon rises vertically at a rate of 15 ft/s.
To find the rate of change of the distance between the motorcycle and the balloon, we need to consider their relative positions.
Let's first find the distance the motorcycle travels in 5 seconds. Since the motorcycle is traveling at a constant speed of 88 ft/s, we can calculate the distance using the formula:
Distance = Speed * Time
Distance = 88 ft/s * 5 s
Distance = 440 ft
After 5 seconds, the motorcycle will have traveled 440 ft.
Now, let's find the height of the balloon after 5 seconds. The balloon rises at a rate of 15 ft/s, so the change in height can be calculated using the formula:
Change in Height = Rate of Change * Time
Change in Height = 15 ft/s * 5 s
Change in Height = 75 ft
After 5 seconds, the balloon will have risen 75 ft.
To determine the rate of change of the distance between the motorcycle and the balloon 5 seconds later, we need to consider the triangle formed by the motorcycle, the balloon, and the ground. The distance between the motorcycle and the balloon is the hypotenuse of this triangle.
Using the Pythagorean theorem, we can calculate the distance between the motorcycle and the balloon after 5 seconds:
Distance^2 = (Distance traveled by motorcycle)^2 + (Change in height)^2
Distance^2 = 440 ft^2 + 75 ft^2
Distance^2 = 193,600 ft^2 + 5,625 ft^2
Distance^2 = 199,225 ft^2
Distance = √199,225 ft
Distance ≈ 446.4 ft
Therefore, the rate of change of the distance between the motorcycle and the balloon 5 seconds later is the derivative of the distance formula with respect to time:
Rate of Change = d(Distance)/dt
To calculate this derivative, we differentiate the equation for the distance between the motorcycle and the balloon:
Rate of Change = d(√(440 ft^2 + 75 ft^2))/dt
Rate of Change = d(√(193,600 ft^2 + 5,625 ft^2))/dt
Evaluating this derivative will give us the rate of change of the distance between the motorcycle and the balloon after 5 seconds. The result will be expressed in ft/s.