A hot-air balloon is 180 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 9 ft/s, what is the rate of change of the distance between the motorcycle and the balloon 9 seconds later?
at time t, the distance z is
z^2 = (180+9t)^2 + (66t)^2 = 4437t^2 + 3240t + 32400
z(9) = 648.81
2z dz/dt = 8874t + 3240
Now plug in z and t to find dz/dt
horizontal distance = x
height = h
z is the distance between,
right triangle so
z^2=h^2+x^2
2 z dz = 2 h dh + 2 x dx
and thus
z dz/dt = h dh/dt + x dx/dt
plug and chug
cool oobleck, never seen it done that way before :)
Charity was probably taught along these lines, same answer as oobleck
At a time of t seconds after the bike passed directly beneath the balloon,
let the height of the balloon be 180+9t ft
let the distance covered by the bike be 66t ft
you have a right-angled triangle.
Let the distance between balloon and bike be d ft (the hypotenuse)
(180+9t)^2 + (66t)^2 = d^2
differentiate with respect to t
2(180+9t)(9) + 2(66t)(66) = 2d dd/dt
9(180+9t) + 4356t = d dd/dt
when t = 9
(180+81)^2 + (594)^2 = d^2
d = √420957
into 9(180+9t) + 4356t = d dd/dt
9(180+81) + 4356(9) = √420957 dd/dt
solve for dd/dt, I got appr 64 ft/sec
check my work
To find the rate of change of the distance between the motorcycle and the balloon, we need to consider the vertical and horizontal motions separately.
Vertical motion:
The balloon rises vertically at a rate of 9 ft/s. So after 9 seconds, the balloon will be 9 ft/s * 9 s = 81 ft higher above the ground. Therefore, the new height of the balloon will be 180 ft + 81 ft = 261 ft.
Horizontal motion:
The motorcycle is traveling on a horizontal road at a constant speed of 66 ft/s. Therefore, after 9 seconds, the motorcycle will have traveled a horizontal distance of 66 ft/s * 9 s = 594 ft.
Now, we can find the distance between the motorcycle and the balloon after 9 seconds using the Pythagorean theorem:
Distance = √(horizontal distance^2 + vertical distance^2)
Distance = √(594 ft^2 + (261 ft - 180 ft)^2)
Distance = √(594^2 + 81^2)
Distance = √(352836 + 6561)
Distance = √359397
Distance ≈ 599.49 ft
Now, the rate of change of the distance between the motorcycle and the balloon can be calculated by finding the derivative of the distance function with respect to time.
d(distance)/dt = (d(horizontal distance)/dt * horizontal distance + d(vertical distance)/dt * vertical distance) / distance
The horizontal distance remains constant over time, so its derivative is 0.
d(horizontal distance)/dt = 0
The vertical distance is increasing at a rate of 9 ft/s.
d(vertical distance)/dt = 9 ft/s
Substituting the values into the formula:
rate of change of distance = (0 * 599.49 ft + 9 ft/s * 599.49 ft) / 599.49 ft
rate of change of distance = (0 + 5395.41 ft/s) / 599.49 ft
rate of change of distance ≈ 9 ft/s
Therefore, the rate of change of the distance between the motorcycle and the balloon 9 seconds later is approximately 9 ft/s.