Quarry

[Numb3rs]

I was given this problem in class today

A Land Developer has been asked by the local council to fence off a disused stone quarry to prevent people from falling into it. The Fence has to be exactly 30 metres away from the edge of the quarry at all times.

the aim is to find the total length of the fence
thanks in advanced

 

[soroban]

Hello, Numb3rs!

A Land Developer has been asked by the local council to fence off
a disused stone quarry to prevent people from falling into it.
The fence has to be exactly 30 metres away from the edge of the quarry at all times.
Find the total length of the fence.

                      *
                    *   *
             100  *       *  100
                *           *
              * 45         45 *
            *   *   *   *   *   *
                      x

We have an isosceles right triangle with legs 100 m.
Hence, the hypotenuse is: .\(\displaystyle x \:=\:100\sqrt{2}\)


A fence is erected exactly 30 feet from the quarry.
On the left and right, there are \(\displaystyle 30 \times 100\) rectangles.
On the bottom is a \(\displaystyle 30 \times 100\sqrt{2}\) rectangle.

                     *.*
                  *:::::::*
                *   *:::*   *
        100   *       *       *  100   
            *       *   *       *
          *       *       *       * 
        *       *           *       *
       *::*   *               *   *::*
       *::::*   *   *   *   *   *::::*
        *:::*                   *:::*
            *   *   *   *   *   *
                      x

\(\displaystyle \text{At each vertex, there is an arc of a circle with radius 30 m.}\)

\(\displaystyle \text{The circumference of this circle is: }\:2\pi(30) \:=\:60\pi\text{ m}\)

\(\displaystyle \text{At the top, the arc has a central angle of }90^o.\)
\(\displaystyle \text{At the lower-left and lower-right, the arcs have central angle of }135^o.\)

\(\displaystyle \text{Since }90^o + 135^o + 135^o \:=\:360^o,\text{ the three arcs comprise a whole circle.}\)
\(\displaystyle \text{The total length of the curved fencing is: }60\pi\text{ m.}\)

\(\displaystyle \text{The total length of the straight fencing is: }\:100 + 100 + 100\sqrt{2} \:=\:200 + 100\sqrt{2}\text{ m.}\)


\(\displaystyle \text{Therefore, the total fencing is: }\;60\pi + 200 + 100\sqrt{2} \;\approx\;529.9\text{ m.}\)

 

[Numb3rs]

thanks heaps soroban, it really helped me