Word Problem: finding area of track, length of lanes

[Hockeyman]

Running around the inside edge of a race track is 400m. Each lane is 1m wide. The turns of the race track are perfect semicircles. The length of the straightaways are 100m long. There are ten lanes in total on the race track.

What is the area of the track?

What is the total length of all the lines?

I'm having difficulty trying to figure this one out. If someone could just get me started on the right track i'm sure i could finish it. Thank you.

 

[galactus]

Here's a rather sloppy drawing, but I hope it helps. Always make a drawing and label the known dimensions.

The area of the entire track can be derived from the info you're given with regards to the inside and the widths of the lanes.

If the sides are 100 and the entire perimeter of the inside is 400, then the semicircles must be of length 100 each.

Use the circle formulas.

Circumference of semicircle: \(\displaystyle \L\\100={\pi}r\rightarrow{r}=\frac{100}{\pi}\)

Now, add 10 to that for the radius of the outside and use the area of a circle formula to find the area of the outside semicircles:

\(\displaystyle \L\\{\pi}(\frac{100}{\pi}+10)^{2}\)

The area of the outside rectangular area:

\(\displaystyle \L\\100(\frac{200}{\pi}+20)\)

Add together:\(\displaystyle \L\\{\pi}(\frac{100}{\pi}+10)^{2}+100(\frac{200}{\pi}+20)=\frac{100({\pi}+10)({\pi}+30)}{\pi}=13863.46 m^{2}\)

Check me out, easy to overlook something and make a mistake.

 

[Hockeyman]

i'm still kind of confused about the whole thing. Could you mabe explain it again, i'm not getting it.

 

[soroban]

Re: Word Problem

Hello, Hockeyman!

A challenging problem!
\(\displaystyle \;\;\)I had to make a good sketch and baby-talk my way through it.
I'll do part (a) for now . . .

Running around the inside edge of a race track is 400m.
Each lane is 1m wide. The turns of the race track are perfect semicircles.
The length of the straightaways are 100m long. There are ten lanes in total on the race track.

(a) What is the area of the track?

(b)What is the total length of all the lines?

This diagram shows the inside edge of the track.

                        100
            * * - - - - - - - - - - * *
         *    |                     |    *
       *      |                     |r     *
              |                     |
      *       |                     |       *
      *       +                     +       *
      *       |                     |       *
              |                     |r
       *      |                     |      *
         *    |                     |    *
            * * - - - - - - - - - - *
                        100

The semicircles have radius \(\displaystyle r\).

The total length of the inside edge is: \(\displaystyle 2\,\times\,100\)
\(\displaystyle \;\;\)plus the circumference of the two semicircles: \(\displaystyle \,2\pi r\)

So we have: \(\displaystyle \,200\,+\,2\pi r\:=\:400\;\;\Rightarrow\;\;r\,=\,\frac{100}{\pi}\)


I don't want to type a new diagram (with the 10 lanes).
\(\displaystyle \;\;\)I hope I can describe the situation clearly enough.

There are 10 lanes, each 1 m wide, running around the region drawn above.

On the two straightaways, there are are two \(\displaystyle 10\times100\) rectangles.
\(\displaystyle \;\;\)Area: \(\displaystyle \,2\times 1000\:=\:2000\) m²

Around the semicircular parts of the track, there are larger semicircles, 10 m wider.
The radius of the larger semicircle is: \(\displaystyle \,\frac{100}{\pi}\,+\,10\) m.

The area of the two larger semicircles is: \(\displaystyle \,\pi\left(\frac{100}{\pi}\,+\,10\right)^2\)
The area of the two smaller semicircles is: \(\displaystyle \,\pi\left(\frac{100}{\pi}\right)^2\)

Hence, the area of the curved track is: \(\displaystyle \,\pi\left(\frac{100}{\pi}\,+\,10\right)^2\,-\,\pi\left(\frac{100}{\pi}\right)^2 \;=\;2000\,+\,100\pi\) m².


(a) Therefore, the total area of the track is: \(\displaystyle \,2000\,+\,(2000\,+\,100\pi)\;=\;4000\,+\,100\pi\) m².

 

[galactus]

Our areas agree, Soroban . I included the center region.

 

[Hockeyman]

Thank you so much Soroban. So then would the total length of the lines be 400m times 10 because there is 10 lanes?

 

[soroban]

Hello, galactus!

Our areas agree, Soroban . I included the center region.

I did, too, in my first run-through . . .


Hokceyman, the total length of all the tracks a bit more complicated.

Suppose the runner in the first lane runs along that inside edge.
His distance is the two straightaways and the two semicircles:
\(\displaystyle \;\;(2\,\times\,100)\,+\,2\pi\left(\frac{100}{\pi}\right) \:= \: 400\)

Suppose the runner in the second lane runs along his inside edge.
His distance is the two straightaways (200 m) plus the slightly larger semicircles.
\(\displaystyle \;\;\)The radius is: \(\displaystyle \,\frac{100}{\pi}\,+\,1.\;\) The curved distance is: \(\displaystyle \,2\pi\left(\frac{100}{\pi}\,+\,1\right) \:=\:200\,+\,2\pi\)
His total distance is: \(\displaystyle \,200\,+\,(200\,+\,2\pi)\:=\:400\,+\,2\pi\)

Suppose the runner in the third lane runs along his inside edge.
His distance is the two straightaways (200 ) plus the slightly larger semicircles.
\(\displaystyle \;\;\)The radius is: \(\displaystyle \,\frac{100}{\pi}\,+\,2.\;\) The curved distance is: \(\displaystyle \,2\pi\left(\frac{100}{\pi}\,+\,2\right) \:=\:200\,+\,4\pi\)
His total distance is: \(\displaystyle \,200\,+\,(200\,+\,4\pi)\:=\:400\,+\,4\pi\)

We find that each lane is \(\displaystyle 2\pi\) meters longer than the one before.
The ten lanes have lengths: \(\displaystyle \,400,\;400+2\pi,\;400+4\pi,\;400+6\pi,\;\cdots\;400+18\pi\)

This is an Arithmetic Series with first term \(\displaystyle a\,=\,400\), common difference \(\displaystyle d\,=\,2\pi\)
\(\displaystyle \;\;\)and \(\displaystyle n\,=\,10\) terms.

The sum of an Arithmetic Series is: \(\displaystyle \,S_n\;=\;\frac{n}{2}[2a\,+\,d(n-1)]\)

So we have: \(\displaystyle \:S_{10}\;=\;\frac{10}{2}[2\cdot400\,+\,(2\pi)(9) \;=\;4000\,+\,90\pi\) m.