Question

MATHS - CLASS X

AREAS RELATED TO CIRCLES

A horse is tied to a peg at one corner of a square shaped grass field of side 15 m by means of a 5m long rope. Find:
(i) the area of that part of field which the horse can graze
(ii) the increase in the grazing are if the rope were 10 m long instead of 5m.

Answer 1

hey mate

here is your answer

(I)-the area of the part of the field which the horse can graze=area of the quadrant whose radius is 5m

=>1/4*π*r²

=>1/4*π*5²

=>1/4*π*25 m²

in order to find the increase in the area ,we first need to find the area of the grazing field when the radio of the quadrant is 10m

=>1/4*π*r²

=>1/4*π*10²

=>1/4*π*100

=>25π

thus,the increase in the area of the grazing field=

=>25π-25π/4

=>3/4*25*π

=>18.75π

Answer 2

Here's the answer

✔️ (Refer the first attachment for question 1)

✔️ (Refer the second attachment for question 2)

In Question 1 ,

See first attachment

Let ABCD be the square shaped field

Length of the rope is 5 cm

Let BG = r = 5 cm

Note that Angle GBH = 90° ( All angles of square are right angled )

So ,

According to the first question,

To find the area of that part of field which the horse can graze ,

We need to find area of sector GBH.

Area of sector GBH = $\sf { \frac{ \theta}{360} \times \pi \: r {}^{2}}$

$\sf{\therefore \ \frac{90}{360} \times 3.14 \times 5 {}^{2}}$

$\sf{ \frac{1}{4} \times 3.14 \times 5 \times 5}$

$\sf{ = \frac{78.5}{4}}$

$\sf {= 19.625}$

Therefore ,

Area of the field where Horse can gaze is 19.625 sq. m.

Now,

Coming to your next question,

Question 2 -

=> Instead of 5 cm , rope is of 15 m

So , Rope's size was increased by 10 m

Now, Refer the second attachment

To find the area gazed by horse,

Find Area of sector TBS

Area of sector TBS $\sf{ = \frac{ \theta}{360} \times \pi \: r {}^{2}}$

$\sf{ = \frac{90}{360} \times 3.14 \times 10 \times 10}$

$\sf{ = \frac{1}{4} \times 314}$

$\sf{= 78.5 \: m {}^{2}}$

Therefore,

Area gazed by horse is 78.5 sq. m now.

Therefore,

Area gazed now - Area gazed earlier = Increase in the area for grazing

= Area of sector GBH - Area of TBS =

= 78.5 - 19.625

= 58.875 sq.m

Thus,

Increase in the area was 58.875 sq.m

Hence, Final answer =>

Question 1's answer
=> Area of the field where Horse can gaze is 19.625 sq. m.

Question 2's answer
=> Increase in the area was 58.875 sq.m

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Thanks!!

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Hope it helps :)

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