Question

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Answer 1

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Answer 2

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Radius of base of a cylinder is 12cm and its height its 9cm,find its covered area and total surface area,(π=3.14)

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$282 6/7 cm^2$

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(See the Fig For better explanation)

The total surface area of a cone is the sum of the area of the base and the lateral area:

$SA=\pi rs^2+\pi rs$

Where:

r = radius

s= slant height

To find the slant height consider the right triangle formed by the radius(base), the height(vertical leg) then the slant height is the hypotenuse so:

$s^2=r^2+h^2=5^2+12^2=169$

$s=13cm$

Thus:

$SA=22/7(25+5*13)=1980/7=282 6/7 cm^2$

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$(T S A =282 (6/7)cm^2$

"Cone Total surface area $" T S A = (1/2) (2pir) * h_s + pi r^2$

$r = 5 cm, h = 12 cm, h_s = sqrt(5^2 + 12^2) = 13 cm$

$(T S A = pi * 5 * 13 + pi 5^2 = 90pi = 1980/7 = 282 (6/7)cm^2$

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cos 60 degree x cos 45 degree x sin 30 degree

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$2-\sqrt{3}$

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to obtain the exact value for the expression use the

trigonometric identities;

$(x)cosC-cosD=-2sin((C+D)/2)sin((C-D)/2)$

$(x)sinC+sinD=2sin((C+D)/2)cos((C-D)/2)$

$(x)sin(x+-y)=sinxcosy+-cosxsiny$

$(x)cos(x+-y)=cosxcosy∓sinxsiny$

simplifying the numerator

"with "C=45" and "D=75

cos45-cos75=-2sin60sin(-15)=2sin60sin15

to obtain the exact value for  sin 15

sin15=sin(45-30)=sin45cos30-cos45sin30

$(sin15)=1/\sqrt2xx\sqrt3/2-1/\sqrt2xx1/2$

$(sin15)=(\sqrt3-1)/(2\sqrt2)=1/4(\sqrt6-\sqrt2)$

$rArrcos45-cos75=2xx\sqrt3/2xx1/4(\sqrt6-sqrt2)$

$=1/4\sqrt3(\sqrt6-\sqrt2)$

simplifying the denominator

$sin45+sin75=2sin60cos(-15)=2sin60cos15$

to obtain the exact value for "cos15

cos15=cos(45-30)=cos 45 cos 30+sin 45 sin 30

$(cos15)=1/sqrt2xx\sqrt3/2+1/\sqrt2xx1/2$

$\(cos15)=(\sqrt3+1)/(2\sqrt2)=1/4\sqrt3(\sqrt6+\sqrt2)$

$rArrsin45+sin75=2xx\sqrt3/2xx1/4(\sqrt6+\sqrt2)$

$=1/4\sqrt3(\sqrt6+\sqrt2)$

$rArr(cos45-cos75)/(sin45+sin75)$

$=(cancel(1/4\sqrt3)(\sqrt6-\sqrt2))/(cancel(1/4\sqrt3)(\sqrt6+\sqrt2))$

$=(\sqrt6-\sqrt2)/(\sqrt6+\sqrt2)xx(\sqrt6-\sqrt2)/(\sqrt6-sqrt2)$  "rationalise denominator

$=(6-2\sqrt12+2)/(6-2)=(8-2(2\sqrt3))/4=2-\sqrt3$