A 15-meter-long rope is hanging vertically from a ceiling. A mouse is climbing up the rope at a constant rate of 2 meters per minute. How fast is the mouse moving when it is 9 meters from the floor?
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To solve this problem, we can use the Pythagorean theorem to find the rate of the mouse's vertical movement. Let's call the rate of the mouse's vertical movement "V" and the rate of the mouse's horizontal movement "H". Since the mouse is climbing vertically, the horizontal movement is zero.
We can create a right triangle with the rope as the hypotenuse and the horizontal and vertical distances as the legs of the triangle. The vertical distance is decreasing by 2 meters per minute, so the rate of change of the vertical distance is -2 m/min.
Using the Pythagorean theorem, we have:
(15)^2 = (H)^2 + (9)^2
225 = H^2 + 81
H^2 = 225 - 81
H^2 = 144
H = 12
Now, we can use the chain rule to find the rate of change of the total distance between the mouse and the floor.
Rate of change of total distance = Rate of change of vertical distance
V = -2 m/min
Now, we can find the rate at which the mouse is moving when it is 9 meters from the floor:
Rate of change of total distance = V = -2 m/min
Therefore, the mouse is moving up the rope at a rate of 2 meters per minute when it is 9 meters from the floor.
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