A boat is pulled into a dock by means of a rope attached to a pulley on the dock. The rope is attached to the front of the boat, which is 7 feet below the level of the pulley. If the rope is pulled through the pulley at a rate of 18 ft/min, at what rate will the boat be approaching the dock when 110 ft of rope is out?
To find the rate at which the boat is approaching the dock, we can use related rates - a concept in calculus.
Let's assign variables to the given values:
- Let x represent the distance between the boat and the dock. This is the distance we want to find (rate of change of x).
- Let y represent the length of the rope that is out.
- Let t represent time.
Based on the given information, we know that:
- The rate at which the rope is pulled through the pulley (dy/dt) is given as 18 ft/min.
- We are given that y = 110 ft when we want to find the rate of change of x.
To find the relationship between x, y, and t, we can use the Pythagorean theorem, which states that in a right triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
In our case, the boat, dock, and the rope form a right triangle. The hypotenuse (c) is the rope length (y), and the other two sides are the distance between the boat and the pulley (x) and the vertical distance between the boat and the level of the pulley (7 ft).
Applying the Pythagorean theorem, we have:
y^2 = x^2 + 7^2
y^2 = x^2 + 49
Now, we differentiate both sides of the equation with respect to t (time):
2y(dy/dt) = 2x(dx/dt)
Since we want to find dx/dt (rate of change of x), we can rearrange the equation:
(dx/dt) = (y(dy/dt)) / x
Substituting the given values:
(dy/dt) = 18 ft/min
y = 110 ft
The equation becomes:
(dx/dt) = (110 ft * 18 ft/min) / x
To find x, we can substitute it back into the Pythagorean theorem equation:
110^2 = x^2 + 49
12100 = x^2 + 49
x^2 = 12100 - 49
x^2 = 12051
x = sqrt(12051)
Finally, we can substitute x into the rate equation to find the rate at which the boat is approaching the dock:
(dx/dt) = (110 ft * 18 ft/min) / sqrt(12051)
Evaluate the expression to find the rate of change of x.