Question

Please solve this question...​

Answer 1

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

• The circumference of two concentric circles are 154 cm and 66 cm.

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

• The area of ring (enclosed by the two circles).

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

$\underline{\boxed{\sf{{Perimeter}_{(Circle)}=2\pi r\:units}}}$

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

$\:\:\:\:\:\:\:\:\:\:\:\:\bullet{\sf{\:{R}^{2}={(Outer\:radius)}^{2}}}$

$\:\:\:\:\:\:\:\:\:\:\:\:\bullet{\sf{\:{r}^{2}={(Inner\:radius)}^{2}}}$

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

First we've to find the radius two circles. It can be found by substituting the given measures in the formula of perimeter of circle. Then, once we obtain the measures of radius, we shall find the area of ring by using the formula of area of annulus (circular ring). Let's do it !!

Outer radius (R):

$\:\:\:\:\:\:\:\:\implies{\sf{P=2\pi R\:units\:\:\:(\pi=\frac{22}{7})}}$

We're given with the measure of circumference of the circle. So,

$\:\:\:\:\:\:\:\:\implies{\sf{154=2\times \frac{22}{7} \times R}}$

$\:\:\:\:\:\:\:\:\implies{\sf{154=\frac{44}{7}\times R}}$

$\:\:\:\:\:\:\:\:\implies{\sf{154\times 7=44\times R}}$

$\:\:\:\:\:\:\:\:\implies{\sf{1078=44\times R}}$

$\:\:\:\:\:\:\:\:\implies{\sf{\frac{1078}{44}=R}}$

$\:\:\:\:\:\:\:\:\implies{\underline{\underline{\sf{24.5\:cm= R}}}}$

Inner radius (r):

Here also, we are given with the measure of circumference of the circle. So,

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

$\:\:\:\:\:\:\:\:\implies{\sf{66=2\times \frac{22}{7}\times r}}$

$\:\:\:\:\:\:\:\:\implies{\sf{66=\frac{44}{7}\times r}}$

$\:\:\:\:\:\:\:\:\implies{\sf{66\times 7=44\times r}}$

$\:\:\:\:\:\:\:\:\implies{\sf{462=44\times r}}$

$\:\:\:\:\:\:\:\:\implies{\sf{\frac{462}{44}=r}}$

$\:\:\:\:\:\:\:\:\implies{\underline{\underline{\sf{10.5\:cm=r}}}}$

Area of ring (annulus):

By substituting the obtained measures in th formula,

$\:\:\:\:\:\:\:\:\implies{\sf{\pi({R}^{2}-{r}^{2})\:sq.units}}$

$\:\:\:\:\:\:\:\:\implies{\sf{\frac{22}{7}\times({24.5}^{2}-{10.5}^{2})}}$

$\:\:\:\:\:\:\:\:\implies{\sf{\frac{22}{7}\times (600.25-110.25)}}$

$\:\:\:\:\:\:\:\:\implies{\sf{\frac{22}{\cancel{7}}\times \cancel{490}}}$

$\:\:\:\:\:\:\:\:\implies{\sf{22\times 70}}$

$\:\:\:\:\:\:\:\:\implies{\underline{\underline{\sf{\red{1540\:{cm}^{2}}}}}}$

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

• Thus, the area of the ring formed by two concentric circles is 1540 cm².

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