Calculus Related Rates(1)

A common calculus one related rates problem.
A person on a dock is pulling in a boat by means of a rope attached to the bow of the boat 1 meter above the water level and passing through a simple pulley located on the dock 8 meters above water level. If he pulls in the rope at a rate of 2 m/s, how fast is the boat approaching the dock when the bow of the boat is 25 meters from a point that is directly below the pulley?

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• common
• calculus
• one
• related
• rates
• problem.
• Math
• Mathematics
• Mathematics Education

[apiersall]

IMO, dz/dt should be positive since it is being pulled in the positive x and y directions (up and rightwards). This would give a positive answer (+2.077 m/s), which answers the original problem more accurately.

"How fast is the boat approaching the dock?" -- it is in fact approaching at positive 2.077 m/s, not negative.

[neiljodymath]

I disagree about dz/dt being positive since z is the length of the hypotenuse of the right triangle used to solve this problem and that distance is not increasing...

I agree, "fast" sort of implies "speed" ...but the answer was listed as negative in the student's answer key who submitted this question, so...

[pickedupapencil]

It should be negative. The person who wrote that is not realizing that the cartesian plane is flipped. Your triangle is backwards according to convention. Rightwards in your example is not positive. The -2.077 m/s tells you that x is decreasing at a rate of 2.077 m/s. So if someone asked for a speed, its 2.077 m/s, but if they ask for a velocity, which tells you more accurately what's going on, its -2.077 m/s. How fast, in math terms, is pretty ambiguous as to which one its really asking for.

[neiljodymath]

@pickedupapencil I see your point. The problem does mention that the boat is approaching the dock...approach is a funny word in math though as well...

[mozam007]

oh my bad , sorry

[neiljodymath]

not a problem, I dont want to have have wrong stuff up there...thanks for the comments

[FunTimesWithGabbi]

thaaaank youuu so much!!!!!!!!

[neiljodymath]

how's that?

read the problem carefully...

"the bow of the boat is 25 meters from a point that is directly below the pulley"

the bow is where the rope is attached and the bow is 25 m horizontal distance from a point directly below the pulley.

How do you interpret this?

[mozam007]

z= 25;

x= 24

x does not equal 25 . read the problem carefully.