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A rectangle contains a strip of width $1,$ as shown below.  Find the area of the strip.

 

 Apr 30, 2024

Best Answer 

 #1
avatar+9664 
+1

Label the diagram as shown. Let \(\angle FCE = \theta\) and \(EC = x\).

Let H be a point on AC such that FH is perpendicular to AC. Then \(FH = 8\).

Note that \(CE = FD = HA = x\). Then \(CH = 10 - x\)

\(CF = \sqrt{8^2 + (10 - x)^2} = \sqrt{x^2 - 20x + 164}\) by using Pythagoras theorem in \(\triangle CFH\).

 

In \(\triangle ECG\)\(\sin \theta = \dfrac{EG}{EC} = \dfrac1x\).

In \(\triangle FHC\)\(\sin \theta = \dfrac{FH}{CF} = \dfrac8{\sqrt{x^2 -20x+164}}\).

 

So we have \(\dfrac1x = \dfrac8{\sqrt{x^2 - 20x + 164}}\). Squaring gives \(\dfrac1{x^2} = \dfrac{64}{x^2 - 20x + 164}\).

 

Rearranging, 

\(64x^2 = x^2 - 20x + 164\\ 63x^2 + 20x - 164 = 0\\ x = \dfrac{-20 \pm \sqrt{20^2 - 4(63)(-164)}}{2(63)}\\ x = \dfrac{8 \sqrt{163} - 10}{63}\text{ (reject negative root)} \)

 

Hence, the area is \(\dfrac{8x}2 = \dfrac{32 \sqrt{163} - 40}{63}\).

 May 2, 2024
 #1
avatar+9664 
+1
Best Answer

Label the diagram as shown. Let \(\angle FCE = \theta\) and \(EC = x\).

Let H be a point on AC such that FH is perpendicular to AC. Then \(FH = 8\).

Note that \(CE = FD = HA = x\). Then \(CH = 10 - x\)

\(CF = \sqrt{8^2 + (10 - x)^2} = \sqrt{x^2 - 20x + 164}\) by using Pythagoras theorem in \(\triangle CFH\).

 

In \(\triangle ECG\)\(\sin \theta = \dfrac{EG}{EC} = \dfrac1x\).

In \(\triangle FHC\)\(\sin \theta = \dfrac{FH}{CF} = \dfrac8{\sqrt{x^2 -20x+164}}\).

 

So we have \(\dfrac1x = \dfrac8{\sqrt{x^2 - 20x + 164}}\). Squaring gives \(\dfrac1{x^2} = \dfrac{64}{x^2 - 20x + 164}\).

 

Rearranging, 

\(64x^2 = x^2 - 20x + 164\\ 63x^2 + 20x - 164 = 0\\ x = \dfrac{-20 \pm \sqrt{20^2 - 4(63)(-164)}}{2(63)}\\ x = \dfrac{8 \sqrt{163} - 10}{63}\text{ (reject negative root)} \)

 

Hence, the area is \(\dfrac{8x}2 = \dfrac{32 \sqrt{163} - 40}{63}\).

MaxWong May 2, 2024
 #2
avatar+128772 
0

Very nice, Max  ....this one is  somewhat difficult  !!!

 

Could you explain how the area is  8x / 2  ???......my small mind doesn't understand (LOL!!!)

 

cool cool cool

CPhill  May 2, 2024

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