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title: 'Math 181 Miniproject 8: Applied Optimization'

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Math 181 Miniproject 8: Applied Optimization.md
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tags: MATH 181
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Math 181 Miniproject 8: Applied Optimization
===

**Overview:**   This miniproject focuses on a central application of calculus, namely *applied optimization*. These problems augment and extend the kinds of problems you have worked with in WeBWorK and class discussions.

**Prerequisites:** Section 3.4 of *Active Calculus.*

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:::info
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.** Two vertical poles of heights 60 ft and 80 ft stand on level ground, with their bases 100 ft apart. A cable that is stretched from the top of one pole to some point on the ground between the poles, and then to the top of the other pole. What is the minimum possible length of cable required? Justify your answer completely using calculus.
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![](https://i.imgur.com/991hAC4.png)
![](https://i.imgur.com/g66pNm7.png)
![](https://i.imgur.com/rEFLpla.png)
![](https://i.imgur.com/ldSfVw1.png)



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**Problem 2.** Use calculus to find the point $(x,y)$ on the parabola traced out by $y = x^2$ that is closest to the point $(3,0)$. 
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![](https://i.imgur.com/xHdddzJ.png)
![](https://i.imgur.com/3u9KJLp.png)

![](https://i.imgur.com/8XBGV6N.png)

The closest point (3,0) on the parabola on the graph is the point (1,1).



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**Problem 3.** Use calculus for find the maximum possible area of a right triangle under the curve 
$$
f\left(x\right)=x\left(x-4\right)^4
$$ 
in the first quadrant with one corner at the origin and one side along the $x$-axis.
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![](https://i.imgur.com/qMwUwr0.png)
![](https://i.imgur.com/5tJPHge.png)






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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.



