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title: 'Math 181 Miniproject 9: Related Rates'

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Math 181 Miniproject 9: Related Rates.md
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tags: MATH 181
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Math 181 Miniproject 9: Related Rates
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**Overview:**   This miniproject focuses on a central application of calculus, namely *related rates*. These problems augment and extend the kinds of problems you have worked with in WeBWorK and class discussions.

**Prerequisites:** Section 3.5 of *Active Calculus.*

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For this miniproject, select EXACTLY TWO of the following and give complete and correct solutions that abide by the specifications for student work. Include a labeled picture with each solution.  Full calculus justification of your conclusions is required.

**Problem 1.** A sailboat is sitting at rest near its dock. A rope attached to the bow of the boat is drawn in over a pulley that stands on a post on the end of the dock that is 5 feet higher than the bow. If the rope is being pulled in at a rate of 2 feet per second, how fast is the boat approaching the dock when the length of rope from bow to pulley is 13 feet?
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x^2 + y^2 =z^2
2x dx/dt + 2y dy/dt = 2z dz/dt
x dx/dt + y dy/dt = z dz/dt
dx/dt = 0 
y dy/dt = z dz/dt
dy/dt - z/y dz/dt
dy/dt = 13.93/12 (2) = 2.14 ft/s
The boat is appraoching the dock at 2.14 ft/s


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**Problem 2.** A baseball diamond is a square with sides 90 feet long. Suppose a baseball player is advancing from second to third base at the rate of 24 feet per second, and an umpire is standing on home plate. Let $\theta$ be the angle between the third baseline and the line of sight from the umpire to the runner. How fast is $\theta$ changing when the runner is 30 feet from third base?
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![](https://i.imgur.com/oexHiGo.png)


tan  θ = x/90        
dx/dt = -24
sec^2(θ) (dθ/dt) = (1/90) (dx/dt)
x = 30
sec θ = sqrt(90^2 + 30^2/ 90) 
sec^2 θ= (90^2)+(30^2)/(90^2) = 1+ 1/9 = 10/9
(10/9) (dθ/dt) = 1/90 (-24)
dθ/dt = -24/100 = -6/25 rads/sec



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**Problem 3.**  Point $A$ is 30 miles west of point $B$. At noon a car starts driving South from point $A$ at a rate of 50 mi/h and a car starts driving South from point $B$ at a rate of 70 mi/h. At 2:00 how quickly is the distance between the cars changing?
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#1 #2 only. 




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To submit this assignment click on the Publish button ![Publish button icon](https://i.imgur.com/Qk7vi9V.png). Then copy the url of the final document and submit it in Canvas.



