Answer: ds/dt = 13.6 ft/s
Explanation:
Given that the balloon is is rising vertically to a certain height above the ground, a bicycle pasess under at a constant rate, we can represent this scenario with a right triangle.
Let s be the hypotenuse of the triangle or the distance between the balloon and the bicycle
Let y be the vertical distance of the balloon from the ground
Let x be the horizontal distance between the balloon and the bicycle
The triangle is shown below
According to the pythagorean theorem,
hypotenuse^2 = one leg^2 + other leg^2
s^2 = y^2 + x^2 equation 1
From the information given,
dy/dt = 20 ft/s
dx/dt = 2 ft/s
We would find ds/dt
By differentiating equation 1, we have
2sds/dt = 2ydy/dt + 2xdy/dt
Dividing through by 2, it becomes
sds/dt = ydy/dt + xdy/dt
Divide both sides by s. It becomes
ds/dt = (ydy/dt + xdy/dt)/s equation 2
Given that after 6 seconds, the bicycle passes under the balloon,
y = 6dy/dt
y = 6 * 20
y = 120
x = 3dx/dt = 3 * 2
x = 12
Just when the balloon is 148 ft above the ground,
x = 12 + 148 = 160
Substituting x = 154 and y = 120 into equation 1,
s^2 = 120^2 + 160^2 = 40000
s = √40000
s = 200
Substituting s = 195.23, x = 154 and y = 120 into equation 2, it becomes
ds/dt = (120 * 20 + 160 * 2)/200
ds/dt = 13.6 ft/s
