Derivatives word problems- Motorboat

[wind]

1. A motorboat coasts toward a dock with its engen off. Its distance s, in meters form the doct t seconds after the engine is turned off is given by the function

s(t) = 100(6 - t) / t + 3 , o less than or equal to t less than or equal to 6
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a) How far is the boat from the dock initially

s(0) = 100(6 - 0) / 0 + 3
s(0) = 100(6 ) / 3
s(0) = 200

therefore the the boat is initially 200m far from the dock.

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b) When does the boat hit the dock?

at t=6?

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c) find the derivitive of the functiion, what does it repersent?

s(t) = 600 -100t / t + 3

sprime(t) = (t+3)(-100) - (600-100t)(1) / (t +3)^2
= -100t -300 - 600 + 100t / (t+3)^2
= -900/ (t +3)^2

it represents the rate of change of speed, in meters, inrespect to time in seconds of the boat.

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d) At what speed does the boat strick the dock?

s(6) = 100(6 - 6) / 6 + 3
= 100(0)/ 6 +3
=0/ 9?

I don't think thats right
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e) What was the speed of the boat when the engen was innitially turned off?

I don't know what to do for this
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Can someone explain these questions? Thanks

 

[skeeter]

1. A motorboat coasts toward a dock with its engen off. Its distance s, in meters form the doct t seconds after the engine is turned off is given by the function

you really need to start using proper grouping symbols (parentheses) to make your expressions mathematically clear

s(t) = 100(6 - t) / (t + 3) , 0 less than or equal to t less than or equal to 6
---------

a) How far is the boat from the dock initially

s(0) = 100(6 - 0) / (0 + 3)
s(0) = 100(6 ) / 3
s(0) = 200

therefore the the boat is initially 200m far from the dock.correct

---------

b) When does the boat hit the dock?

at t=6? yes, s(6) = 0

-----------

c) find the derivitive of the functiion, what does it repersent?

s(t) = (600 -100t)/(t + 3)

sprime(t) = (t+3)(-100) - (600-100t)(1) / (t +3)^2
= -100t -300 - 600 + 100t / (t+3)^2
= -900/ (t +3)^2

it represents the rate of change of position, which is velocity, in meters per second.
-----------

d) At what speed does the boat strick the dock?

s(6) = 100(6 - 6) / 6 + 3
= 100(0)/ 6 +3
=0/ 9?

I don't think thats right
velocity at t = 6 is s'(6), not s(6). speed is the magnitude of velocity.
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e) What was the speed of the boat when the engen was innitially turned off?

I don't know what to do for this |s'(0)| = speed at time t = 0
-------

Can someone explain these questions? Thanks

 

[wind]

d) At what speed does the boat strick the dock?

s(6) = 100(6 - 6) / 6 + 3
= 100(0)/ 6 +3
=0/ 9?

I don't think thats right
velocity at t = 6 is s'(6), not s(6). speed is the magnitude of velocity.
-----------

e) What was the speed of the boat when the engen was innitially turned off?

I don't know what to do for this |s'(0)| = speed at time t = 0

Thanks skeeter,

So

d)

-900 / (t + 3) ^2
=-900 / (6 + 3) ^2
=-900 / 81
=11.1m/s

e)

-900 / (0 + 3) ^2
-900 / 9
-100 m/s

Is that right?

 

[skeeter]

d. correct

e. speed is always (+).

whoever made up this problem has no realistic concept of motion ... 100 m/s is mighty fast for a boat. (100 m/s is approx 224 miles per hour)

 

[wind]

thanks, skeeter