Area - Study Mode

[#191] In the given diagram, ABCD is a square and semi-circular regions have been added to it by drawing two semi-circles with AB and CD as diameters. If the total area of the three regions is 350 sq.cm, then the length of the side of the square is equal to :
Correct Answer

(D) 14 cm

Explanation

Solution: Let the length of the side of the square be x cm Then, radius of each semi-circle = $$left( {frac{x}{2}}
ight)$$ cm Total area : $$eqalign{
& = left[ {{x^2} + frac{pi }{2}{{left( {frac{x}{2}}
ight)}^2} + frac{pi }{2}{{left( {frac{x}{2}}
ight)}^2}}
ight]c{m^2} cr
& = left( {{x^2} + frac{pi }{4} imes {x^2}}
ight)c{m^2} cr} $$ $$eqalign{
& herefore {x^2} + frac{pi }{4} imes {x^2} = 350 cr
& Rightarrow {x^2} + frac{{22{x^2}}}{{28}} = 350 cr
& Rightarrow {x^2} + frac{{11{x^2}}}{{14}} = 350 cr
& Rightarrow frac{{25{x^2}}}{{14}} = 350 cr
& Rightarrow {x^2} = left( {frac{{350 imes 14}}{{25}}}
ight) cr
& Rightarrow {x^2} = 196 cr
& Rightarrow x = 14,cm cr} $$

[#192] The radius of the circum-circle of an equilateral triangle of side 12 cm is :
Correct Answer

(D) $$4sqrt 3 $$

Explanation

Solution: Radius of circum-circle : $$eqalign{
& = frac{a}{{sqrt 3 }} cr
& = frac{{12}}{{sqrt 3 }},cm cr
& = 4sqrt 3 ,cm cr} $$

[#193] Three circle of radius 3.5 cm are placed in such a way that each circle touches the other two. The area of the portion enclosed by the circles is :
Correct Answer

(A) 1.967 cm 2

Explanation

Solution: Required area = (Area of an equilateral Δ of side 7 cm) - (3 × Area of sector with θ
= 60° and r = 3.5 cm) $$ = left[ {left( {frac{{sqrt 3 }}{4} imes 7 imes 7}
ight) - left( {3 imes frac{{22}}{7} imes 3.5 imes 3.5 imes frac{{60}}{{360}}}
ight)}
ight]c{m^2}$$ $$eqalign{
& = left( {frac{{49sqrt 3 }}{4} - 11 imes 0.5 imes 3.5}
ight)c{m^2} cr
& = (21.217 - 19.25)c{m^2} cr
& = 1.967,c{m^2} cr} $$

[#194] The height of a triangle is equal to the perimeter of a square whose diagonal is $$8sqrt 2 $$ metre and the base of the same triangle is equal to the side of a square whose area is 729 sq.metre. What is the area of the triangle ? (in sq. metre)
Correct Answer

(D) 432 sq.metre

Explanation

Solution: Height of triangle = perimeter of square Diagonal of square = $$8sqrt 2 $$ m ∴ Length of each side of square : $$ = frac{{8sqrt 2 }}{{sqrt 2 }} = 8,m$$ ∴ Perimeter of square = 4 × 8 = 32 m = Height Area of other square = 729 Side of square = $$sqrt {729} $$ xa0= 27 m = base of triangle ∴ Area of triangle : $$eqalign{
& = frac{1}{2} imes { ext{Base}} imes { ext{Height}} cr
& = frac{1}{2} imes 27 imes 32 cr
& = 432{ ext{ sq}}{ ext{.metre}} cr} $$

[#195] The ratio of length and breadth of a rectangle is 3 : 2 respectively. The respective ratio of its perimeter and area is 5 : 9. What is the breadth of the rectangle in metres ?
Correct Answer

(A) 6 m

Explanation

Solution: Let the length and breadth of the rectangle be 3x and 2x respectively Then, Perimeter = 2 (3x + 2x) = 10x And, Area = (3x × 2x) = 6x 2 $$eqalign{
& herefore frac{{10x}}{{6{x^2}}} = frac{5}{9} cr
& Rightarrow 30x = 90 cr
& Rightarrow x = 3 cr} $$ So, breadth = (2 × 3) m = 6 m