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The given figure depicts a racing track whose left and right ends are semicircular. The distance between the two inner parallel line segments is 60 m and they are each 106 m long. If the track is 10 m wide, find:
(i) the distance around the track along its inner edge.
(ii) the area of the track.
Given:
The given figure depicts a racing track whose left and right ends are semicircular. The distance between the two inner parallel line segments is 60 m and they are each 106 m long.
The track is 10 m wide.
To do:
We have to find
(i) the distance around the track along its inner edge.
(ii) the area of the track.
Solution:
(i) Circumference of the semicircular ends at both sides $=\frac{2 \pi r}{2}$
$=\pi r$
Radius of the circular ends on both sides $r=\frac{60}{2}$
$=30 \mathrm{~cm}$
Circumference of the semicircular ends $=\frac{22}{7} \times 30$
$=\frac{660}{7} \mathrm{~cm}$
Total length of the semicircular ends $=\frac{2 \times 660}{7}$
$=\frac{1320}{7} \mathrm{~cm}$
Total length of the inner racing track $=106+106+\frac{1320}{7}$
$=212+\frac{1320}{7}$
$=\frac{1484+1320}{7}$
$=\frac{2804}{7} \mathrm{~cm}$
The distance around the track along its inner edge is $\frac{2804}{7} \mathrm{~cm}$.
(ii) Radius of the outer semicircular end $=30+10$
$=40 \mathrm{~cm}$
Area of the outer semicircular ends $=\frac{1}{2}(\pi r^{2})$
$=\frac{1}{2} \times \frac{22}{7} \times 40 \times 40$
$=\frac{11 \times 40 \times 40}{7}$
$=\frac{17600}{7} \mathrm{~cm}^{2}$
Area of the inner semicircular ends $=\frac{1}{2} \times \frac{22}{7} \times 30 \times 30$
$=\frac{11 \times 900}{7}$
$=\frac{9900}{7} \mathrm{~cm}^{2}$
Area of the track between semicircular ends $=\frac{17600}{7}-\frac{9900}{7}$
$=\frac{7700}{7} \mathrm{~cm}^{2}$
$=1100 \mathrm{~cm}^{2}$
Area of the tracks at both semicircular ends $= 2 \times 1100$
$= 2200\ cm^2$
Area of the two rectangular portions $= 2 l \times h$
$= 2 \times 106 \times 10$
$= 2120\ cm^2$
Total area of the track $=$ Area of the track at semicircular ends $+$ Area of the rectangular portions
$= 2200 + 2120$
$= 4320\ cm^2$