A barge whose deck is 5 metres below the level of a dock is being drawn in by
means of a cable attached to the dock and passing through a ring on the dock.
When the barge is 12 metres away and approaching the dock at 3 4
metre per
minute, how fast is the cable being pulled in?
Tan A = 5m/12m, A = 22.6o.
Cos22.6 = 3.4/V, V = 3.68 m/min.
To find the rate at which the cable is being pulled in, we need to use the concept of related rates.
Let's define the variables:
- Let "d" represent the distance between the barge and the dock at any given time.
- Let "h" represent the depth of the barge's deck below the level of the dock at any given time.
- Let "x" represent the horizontal distance between the dock and the point where the cable enters the water (assume the barge is directly below this point).
Given information:
- The deck of the barge is 5 meters below the level of the dock, so h = 5.
- The barge is 12 meters away from the dock, so d = 12.
- The barge is approaching the dock at a rate of 3.4 meters per minute, so dd/dt = 3.4.
We need to find dh/dt, the rate at which the cable is being pulled in. To do this, we need to find an equation that relates the variables d and h.
Using the Pythagorean theorem, we have:
d^2 = x^2 + h^2
Differentiating both sides of the equation with respect to time (t), we get:
2d * dd/dt = 2x * dx/dt + 2h * dh/dt
Since we are interested in finding dh/dt, we rearrange the equation to solve for it:
dh/dt = (d * dd/dt - x * dx/dt) / h
Now, let's find the values for x and dx/dt.
At any given time, we have a right triangle formed by the barge, the dock, and the submerged cable. The horizontal distance x can be found using similar triangles:
x / d = h / (h + 5)
Cross-multiplying and simplifying:
x = (d * h) / (h + 5)
Now, differentiate x with respect to t (time) to get dx/dt:
dx/dt = (d * dh/dt * (h + 5) - d * h * dh/dt) / (h + 5)^2
Plugging in the given values:
dx/dt = (12 * dh/dt * (5 + 5) - 12 * 5 * dh/dt) / (5 + 5)^2 = dh/dt / 2
Now, we can substitute the values of d, dd/dt, x, and dx/dt into the equation for dh/dt:
dh/dt = (12 * 3.4 - (12 * 5 * 3.4) / 2) / 5
= (40.8 - 204) / 10
= -16.8 meters per minute
So, the cable is being pulled in at a rate of 16.8 meters per minute. Note that the negative sign indicates that the barge is sinking as it approaches the dock.