MHB Problem of the Week #118 - June 30th, 2014

[Chris L T521]

Here's this week's problem!

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Problem: Elizabeth has just inherited from her uncle a tract of land whose boundary is described as follows: "From the maple tree in front of the house, go 1000 yards northeast; from that point go 1200 yards northwest; from that point, go 800 yards south; from that point, go back to the maple tree." What is the area of Elizabeth's inheritance?

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Sugestion: The formula in http://mathhelpboards.com/potw-university-students-34/problem-week-36-december-3rd-2012-a-2632.html comes in pretty handy here. (hooray for referencing previous POTWs!) (Smirk)

Remember to read the http://www.mathhelpboards.com/showthread.php?772-Problem-of-the-Week-%28POTW%29-Procedure-and-Guidelines to find out how to http://www.mathhelpboards.com/forms.php?do=form&fid=2!

 

[Chris L T521]

No one answered this week's problem. You can find the solution below.

[sp]Let the origin (0,0) denote the location of the maple tree. Following the directions provided in the will, we see that the points going counterclockwise are $(500\sqrt{2},500\sqrt{2})$, $(-100\sqrt{2},1100\sqrt{2})$, $(-100\sqrt{2},1100\sqrt{2}-800)$. Using the formula in POTW #36, we see that the area of the region is
\[\begin{aligned}A &= \frac{1}{2}\left(\begin{vmatrix}0 & 500\sqrt{2}\\ 0 & 500\sqrt{2}\end{vmatrix} + \begin{vmatrix}500\sqrt{2} & -100\sqrt{2}\\500\sqrt{2} & 1100\sqrt{2}\end{vmatrix} + \begin{vmatrix} -100\sqrt{2} & -100\sqrt{2} \\ 1100\sqrt{2} & 1100\sqrt{2} - 800\end{vmatrix} + \begin{vmatrix}-100\sqrt{2} & 0\\ 1100\sqrt{2}-800 & 0 \end{vmatrix}\right)\\ &= \frac{1}{2}(1300000+80000\sqrt{2})\\ &= 650000+40000\sqrt{2}.\end{aligned}\]
Thus, the area enclosed is $650000+40000\sqrt{2}$ sq. yards.[/sp]