Question and Answers Forum

All Questions      Topic List

Geometry Questions

Previous in All Question      Next in All Question      

Previous in Geometry      Next in Geometry      

Question Number 20944 by tawa tawa last updated on 08/Sep/17

Answered by dioph last updated on 09/Sep/17

Commented by dioph last updated on 09/Sep/17

r((√3)/2) = 7(√3) ⇒ r = 14 cm  A = 4×(((πr^2 )/6)) − 2×(((r^2 (√3))/4))  = 4×(((22×14^2 )/(7×6))) − 2×(((14^2 ×(√3))/4))  = ((1232)/3) − 98(√3)  = (411 − (1/3)) − (√(28812))  But (√(28900)) = 170, and  (170−(1/3))^2 =170^2 −((340)/3)+(1/9) < 170^2 −113 = 27787  Hence (170−(1/3))<(√(28812)) < 170  ⇒ 241−(1/3) < A < 241  So A = 241 cm^2  to the nearest cm^2

$${r}\frac{\sqrt{\mathrm{3}}}{\mathrm{2}}\:=\:\mathrm{7}\sqrt{\mathrm{3}}\:\Rightarrow\:{r}\:=\:\mathrm{14}\:\mathrm{cm} \\ $$$${A}\:=\:\mathrm{4}×\left(\frac{\pi{r}^{\mathrm{2}} }{\mathrm{6}}\right)\:−\:\mathrm{2}×\left(\frac{{r}^{\mathrm{2}} \sqrt{\mathrm{3}}}{\mathrm{4}}\right) \\ $$$$=\:\mathrm{4}×\left(\frac{\mathrm{22}×\mathrm{14}^{\mathrm{2}} }{\mathrm{7}×\mathrm{6}}\right)\:−\:\mathrm{2}×\left(\frac{\mathrm{14}^{\mathrm{2}} ×\sqrt{\mathrm{3}}}{\mathrm{4}}\right) \\ $$$$=\:\frac{\mathrm{1232}}{\mathrm{3}}\:−\:\mathrm{98}\sqrt{\mathrm{3}} \\ $$$$=\:\left(\mathrm{411}\:−\:\frac{\mathrm{1}}{\mathrm{3}}\right)\:−\:\sqrt{\mathrm{28812}} \\ $$$$\mathrm{But}\:\sqrt{\mathrm{28900}}\:=\:\mathrm{170},\:\mathrm{and} \\ $$$$\left(\mathrm{170}−\frac{\mathrm{1}}{\mathrm{3}}\right)^{\mathrm{2}} =\mathrm{170}^{\mathrm{2}} −\frac{\mathrm{340}}{\mathrm{3}}+\frac{\mathrm{1}}{\mathrm{9}}\:<\:\mathrm{170}^{\mathrm{2}} −\mathrm{113}\:=\:\mathrm{27787} \\ $$$$\mathrm{Hence}\:\left(\mathrm{170}−\frac{\mathrm{1}}{\mathrm{3}}\right)<\sqrt{\mathrm{28812}}\:<\:\mathrm{170} \\ $$$$\Rightarrow\:\mathrm{241}−\frac{\mathrm{1}}{\mathrm{3}}\:<\:{A}\:<\:\mathrm{241} \\ $$$$\mathrm{So}\:{A}\:=\:\mathrm{241}\:\mathrm{cm}^{\mathrm{2}} \:\mathrm{to}\:\mathrm{the}\:\mathrm{nearest}\:\mathrm{cm}^{\mathrm{2}} \\ $$

Commented by tawa tawa last updated on 10/Sep/17

God bless you sir

$$\mathrm{God}\:\mathrm{bless}\:\mathrm{you}\:\mathrm{sir} \\ $$

Terms of Service

Privacy Policy

Contact: info@tinkutara.com