A boat is pulled into a dock by a rope attached to the bow of the boat an passing through the pulley on the deck that is 1m high than the bow of the boat ,if the rope is pulled im at the rate of 0.8m/s how fast does the boat approach the deck when it is 10m from the dock?
Do math
To find the speed at which the boat approaches the deck, we can use the concept of similar triangles.
Let's assume that the distance between the deck and the dock is "d" meters.
We know that the rope is pulled in at a rate of 0.8 m/s, so the rate at which the distance between the boat and the dock is changing is also 0.8 m/s.
Now, we have a right triangle formed by the boat, the deck, and the dock. The height of the deck from the bow of the boat is 1 meter more than the distance between the deck and the dock.
Using similar triangles, we can set up the following proportion:
(Height of the deck) / (Distance between the deck and the dock) = (Height of the boat) / (Distance between the bow of the boat and the dock)
Let's denote the height of the deck as h and the distance between the bow of the boat and the dock as x.
The proportion becomes:
h / d = (h + 1) / x
Cross-multiplying, we get:
h * x = (h + 1) * d
Rearranging the equation, we have:
h * x - h * d = d
Factoring out h, we get:
h * (x - d) = d
Finally, solving for h, we have:
h = d / (x - d)
Now, we can differentiate both sides of the equation with respect to time (t) to find how the height of the deck changes as the boat approaches the dock.
dh/dt = (d / (x - d)) * (dx/dt)
Substituting the given values, we have:
dh/dt = (10 / (10 - 10)) * (0.8) [Substituting d = 10 and dx/dt = 0.8]
dh/dt = (10 / 0) * (0.8)
Since we obtained a division by 0, this means that the rate of approach of the boat to the deck is undefined.
In this situation, the boat will never reach the deck because the height of the deck is 1 meter more than the distance between the deck and the dock.