Question

14. A horse has been tied to a peg at A, in the corner of a fenced
garden, with a rope 12.6 m long. At point B, there is another
peg, to which a goat is tied with a rope 4.2 m long. Find
the ratio between the areas they can graze.​

Answer 1

${\underline{\underline{\bf{Given:}}}}$

• That, a horse has been tied into a peg in the corner of a fenced garden with a rope of length 12.6 m.
• And a goat is tied with a rope of length 4.2 m in another peg inside the same garden.

${\underline{\underline{\bf{Need\:to\: find:}}}}$

• The ratio between the areas they (the horse and the goat) can graze.

${\underline{\underline{\bf{Formulae\:to\:be\:used:}}}}$

$\underline{\boxed{\sf{{Area}_{(Quadrant-circle)}=\frac{1}{4}\pi{r}^{2}\:sq.units}}}$

$\underline{\boxed{\sf{{Area}_{(Circle)}=\pi{r}^{2}\:sq.units}}}$

${\underline{\underline{\bf{Solution:}}}}$

Area of region, the horse can graze:

If we look at the image (attachment), we can find that the horse is tied to the corner of the garden. So, the area it can graze will be in the form of quadrant (circle). As per the given data, the area of quadrant can be found by substituting 12.6 m in place of radius in the formula of area of quadrant.

$\:\:\:\:\:\:\:\implies{\sf{\frac{1}{4}\pi{r}^{2}\:sq.units\:\:\:(\pi=\frac{22}{7})}}$

$\:\:\:\:\:\:\:\implies{\sf{\frac{1}{\cancel{4}}\times \frac{\cancel{22}}{7}\times {(12.6)}^{2}}}$

$\:\:\:\:\:\:\:\implies{\sf{\frac{1}{2}\times \frac{11}{7}\times 12.6\times 12.6}}$

$\:\:\:\:\:\:\:\implies{\sf{\frac{1}{2}\times \frac{11}{7}\times 158.76}}$

$\:\:\:\:\:\:\:\implies{\sf{\frac{1}{2}\times 249.48}}$

$\:\:\:\:\:\:\:\implies{\underline{\underline{\sf{124.74\:{m}^{2}}}}}$

Area of region, the goat can graze:

According to the question, if we look at the image, the goat is tied at a point inside the garden. So, the area it can cover will be in the shape for circle. Let's substitute the given measure in the formula of area of circle.

$\:\:\:\:\:\:\:\implies{\sf{\pi{r}^{2}\:sq.units\:\:\:(\pi=\frac{22}{7})}}$

$\:\:\:\:\:\:\:\implies{\sf{\frac{22}{7}\times {(4.2)}^{2}}}$

$\:\:\:\:\:\:\:\implies{\sf{\frac{22}{7}\times 4.2\times 4.2}}$

$\:\:\:\:\:\:\:\implies{\sf{\frac{22}{7}\times 17.64}}$

$\:\:\:\:\:\:\:\implies{\underline{\underline{\sf{55.44\:{m}^{2}}}}}$

Ratio of between the areas they can graze:

$\rightarrow{\sf{{Horse}_{(Area)}:{Goat}_{(Area)}}}$

$\rightarrow{\sf{\underline{124.74}:\underline{55.44}}}$

$\rightarrow{\sf{\underline{62.37}:\underline{27.72}}}$

$\rightarrow{\sf{\red{\underline{8.91}:\underline{3.96}}}}$

${\underline{\underline{\bf{Final\:answer:}}}}$

• Thus, the ratio between the areas they can graze is 8.91 : 3.96.

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