Question

In a cycling race the wheel of a bicycle of the cyclist take total 304 to complete the circuit. The cycling track is around of football ground that is of rectangular shape with semi circular shaped on both side the perimeter of the inner rectangular ground is 496 m. The inner semicircular shape on both side has a length of 154 m each (from A to I to B or from c to l to d). The width of the track is 7 m.

1) What is the area of the football field?
2) What what is the length of the full football ground?
3) What is the radius of the wheel of bicycle?
4) What is the area of cycling track?
5) In how many rounds does cyclist will cover 12 km distance​?

Answer 1

Answer:

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Step-by-step explanation:

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Answer 2

Given:

1. No of rotations by the wheel to complete one round of the ground = 304
2. Perimeter of the rectangular field = 496 m
3. Perimeter of semicircle (each) = 154 m
4. Width of the track = 7 m

To find:

1. Area of the football field.
2. Length of the full football ground.
3. Radius of the wheel of bicycle
4. Area of the cycling track
5. Number of rotations of the bike for travelling 12 km.

Solution:

Step 1

We have been given the perimeter of semicircle on both sides of the rectangular field equal to 154 m.

We know, perimeter of semicircle is πR where R is the radius of the semicircle and the $\frac{1}{2}$ breadth of the rectangular part of the football field.

Hence,

π$R$ $= 154$

$\frac{22}{7}R=154$

$\frac{R}{7} = 7$     $;$ $R=49$

Therefore, $R=49 cm$ is the radius of the semicircle and breadth of the rectangular field is $98 cm.$

Step 2

We have also been given the perimeter of the rectangular field equal to $496m.$ We know perimeter of rectangle is $2(l+b)$ where $l$ and $b$ are the length and breadth of the rectangular field respectively.

$2(l+b)=496\\l+98=248$   $;$ $hence$

$l=150 cm$

Now,

Area of the football field will be = area of the rectangular part with $l=150cm$ and $b=98cm$ + $2$ ×(area of semicircular part with radius 49)

Hence,

Area of the field = $l$ × $b$ $+$  (π $R^{2}$)

Substituting the given values, we get

$A=(150)(98)+\frac{22}{7}(49)(49)$

$A=14700+7546 =22246$

Hence, the area of the football field = $22246m^{2}$

Step 3

The full length of the football ground = length of the rectangular field + 2×(radius of the semicircular field)

Hence,

The length of the football field $=150+2$ × $(49)$

Length $=150+98=248$

Therefore, the length of the complete football field is $248m$

Step 4

We have been given that the wheel of the bicycle makes 304 rotations on completion of riding along the perimeter of the football field.

Hence,

Number of rotations made by the wheel × distance covered by 1 rotation of the wheel (circumference of the wheel) will be equal to the perimeter of the football field.

The perimeter of the football field = (twice the length of the rectangular field) + 2 ×(circumference of the semicircular part)

Perimeter = $2(l)$ $+$ $2($ π $R)$

$= 2(150)+2(\frac{22}{7}(49) )\\= 300+2(154)\\=608m$

Let, radius of wheel be $"r"$

Therefore,

$304$ × $2$ π $r=608$

π $r=1$     $;$  hence

$r=\frac{7}{22}m$

Step 5

The cycling track, has two rectangular shaped field on either side of the length of the field and two semicircle shaped structures across the breadth of the field.

The semicircular field has 2 radii, one inner of the football field $(R_{1} )=49m$ and the other on the edge of the cycling track $(R_{2} )= R_{1}+7=56m$

Hence, the area of the cycling track = π $((R_{2} )^{2}- (R_{1} )^{2} )$

Substituting the given values, we get

$A=\frac{22}{7}(3136-2401)$  

$A=\frac{22}{7}(735)$

$A=2310m^{2}$

Hence, the semicircular region has  $=2310m^{2}$ area.

Now,

Area of the rectangular cycling track =  $l$ × $b$  

Since there are 2 rectangular tracks on either side of the rectangle,

$A=$ $2$ × $l$ × $b$

$A=2(150)(7)\\A=2(1050)$ $;$ $or$

$A=2100m^{2}$

Now, total area of the cycling track = $2310+2100=4410m^{2}$

Step 6

We know, the whole length of the cycling track $=608m$

and

Number of rotations made by the wheel in travelling $608m=304$

Hence,

Number of rotations made by the wheel in travelling $12000m= \frac{(12000)(304)}{608}$

Therefore,

Number of rotations will be $6000.$

Final answer:

1. Area of the football field is 22246 m².
2. Length of the football field is 248 m.
3. Radius of the wheel of the bicycle is $\frac{7}{22}m$.
4. Area of the cycling track is 4410 m².
5. The wheel will take 6000 rotations to travel a distance of 12 km.