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Calculus/word problems.

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Expert: - 6/4/2009


Question
I couldn't figure these out for a hw awhile back, could you help provide the answers with the work and formulas to find these?? it would be very helpful for future tests

A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is 1 m higher than the bow of the boat.

If the rope is pulled in at a rate of 1.2 m/s, how fast is the boat approaching the dock when it is 9 m from the dock?

2. A baseball diamond is a square with sides of length 90 ft. A batter hits the ball and runs toward first base with a speed of 20 ft/s.

At what rate is his distance from second base changing when he is halfway to first base?
Answer = ft/s

At what rate is his distance from third base changing at the same moment?
Answer = ft/s  

Answer

Baseball
A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is 1 m higher than the bow of the boat.  If the rope is pulled in at a rate of 1.2 m/s, how fast is the boat approaching the dock when it is 9 m from the dock?

Let x be the horizontal distance, y be the vertical distance,
and r bet the distance between the boat and the pulley.

We know that x = 9, y = 1, and r² = x²+y².
Also, we know that only x and r are changing, y is a constant.

In the equation, we can say that 2r dr = 2x dx
since y is a constant.

Putting in what x and r have the value of gives us
2√101 dr = 2*9 dx.

We know that dr is the rate at which the rope is being pulled.
Since the rope is being pulled in, the change is negative.
This says that dr = -1.2.

Cancelling the 2's in the equation and dividing by 9 gives us
(√101)(-1.2)/9 = dx.  Evaluate this, and that gives a negative value for dx since x is decreasing.  The question, however, wants the negative of this since it asks for how fast the distance is decreasing.


2. A baseball diamond is a square with sides of length 90 ft. A batter hits the ball and runs toward first base with a speed of 20 ft/s.

We are given a baseball diamond, which is really a square that is tipped 45°.
We know that y = 90, x = 45, dx = -20.

At what rate is his distance from second base changing when he is halfway to first base?
Answer = ft/s

Note that z² = x² + y², so 2z dz = 2x dx since y is a constant.
This means that dz = (x/z) dx.

Now z is known to be √(45²+90²) = 45√(1²+2²) = 45√5.
The value for x was given as 45.
This means that dz = (45/45√5) dx = (1/√5) dx = (√5/5)dx.
We know that dx = -20, so dz = (√5/5)(-20) = -4√5.

At what rate is his distance from third base changing at the same moment?
Answer = ft/s

Looking at the triangle, everything is symmetric between the upper and lower triangle,
so it is a negative of the other one and is therefore 4√5.

Scotto

Expertise

Any kind of calculus question you want. I also have answered some questions in Physics (mass, momentum, falling bodies), Chemistry (charge, reactions, symbols, molecules), and Biology.

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