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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$2c\cdot\frac{dc}{dt}=2b\cdot\frac{db}{dt}$
We know $\frac{dc}{dt}=2$ft/sec when $c=13$ft and $a=5$ft.
We need $\frac{db}{dt}$ when $c=13$ft and $a=5$ft.
$2\left(13\right)\cdot\left(2\right)=2\left(12\right)\cdot\frac{db}{dt}$
$2.1666666667=\frac{db}{dt}$
When the bow is $13$ft away from the pulley, the boat is approaching the dock at a rate of $2.1666666667$ feet per second.

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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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$\frac{d\theta}{dt}=\frac{1}{1+\left(\frac{o}{90}\right)^{2}}\cdot\frac{1}{90}\frac{do}{dt}$
We know $\frac{do}{dt}=24$ft/sec and $o=30$ft.
We need $\frac{d\theta}{dt}$ when $o=30$ft.
$\frac{d\theta}{dt}=\frac{1}{1+\left(\frac{\left(30\right)}{90}\right)^{2}}\cdot\frac{1}{90}\left(24\right)$
$\frac{d\theta}{dt}=0.24$
When the runner is $30$ft from third base, the angle that is made between the third baseline and the umpire's line of sight, $\theta$, is decreasing at a rate of $0.24$ feet per second.

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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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