Area - Study Mode
[#76] The length of a rectangle is three times of its width. If the length of the diagonal is $$8sqrt {10} $$ cm, then the perimeter of the rectangle is :
Correct Answer
(D) 64 cm
Explanation
Solution: Let breadth = x cm Then, length = 3x cm $$eqalign{
& Rightarrow {x^2} + {left( {3x}
ight)^2} = {left( {8sqrt {10} }
ight)^2} cr
& Rightarrow 10{x^2} = 640 cr
& Rightarrow {x^2} = 64 cr
& Rightarrow x = 8 cr} $$ So, length = 24 cm and breadth = 8 cm ∴ Perimeter = [2(24 + 8)] cm = 64 cm
[#77] A typist uses a paper 30 cm by 15 cm. He leaves a margin of 2.5 cm at the top and bottom and 1.25 cm on either side. What percentage of paper area is approximately available for typing ?
Correct Answer
(C) 70%
Explanation
Solution: Area of the sheet = (30 × 15) cm 2 = 450 cm 2 Area used for typing = [(30 - 5) × (15 - 2.5)] cm 2 = 312.5 cm 2 ∴ Required percentage : $$eqalign{
& = left( {frac{{312.5}}{{450}} imes 100}
ight)\% cr
& = 69.4\% approx 70\% cr} $$
[#78] In the given figure, ABC is an equilateral triangle which is inscribed inside a circle and whose radius is r. Which of the following is the area of the triangle ?
Correct Answer
(D) $${left( {r - DE}
ight)^{frac{1}{2}}}{left( {r + DE}
ight)^{frac{3}{2}}}$$
Explanation
Solution: We have : $$eqalign{
& AE x08ot BC{ ext{ and }} cr
& AD = BD = CD = r cr
& AE = AD + DE = r + DE cr} $$ In Δ BDC, $$eqalign{
& BE = sqrt {{{left( {BD}
ight)}^2} - {{left( {DE}
ight)}^2}} cr
& ,,,,,,,,,,, = sqrt {{r^2} - {{left( {DE}
ight)}^2}} cr
& ,,,,,,,,,,, = sqrt {left( {r - DE}
ight)left( {r + DE}
ight)} cr} $$ ∴ Area of the triangle : $$eqalign{
& = frac{1}{2} imes BC imes AE cr
& = frac{1}{2} imes 2BE imes AE cr
& = BE imes AE cr
& = sqrt {left( {r - DE}
ight)left( {r + DE}
ight)} left( {r + DE}
ight) cr
& = {left( {r - DE}
ight)^{frac{1}{2}}}{left( {r + DE}
ight)^{frac{3}{2}}} cr} $$
[#79] A kite-shaped quadrilateral of the largest possible area is cut from a circular sheet of paper. If the lengths of the sides of the kite are in the ratio 3 : 3 : 4 : 4, what percentage of the circular sheet is wasted ?
Correct Answer
(B) 39%
Explanation
Solution: Clearly, the longer diagonal of the kite is the diameter of the circle Also, ∠ABC = 90° (angle in a semi-circle) Let AB = AD = 3x and BC = CD = 4x Then, $$AC = sqrt {A{B^2} + B{C^2}} = 5x$$ Area of the kite = 2 × area (ΔABC) $$eqalign{
& = 2 imes { ext{Area (}}vartriangle { ext{ABC)}} cr
& = 2 imes frac{1}{2} imes BC imes AB cr
& = 3x imes 4x cr
& = 12{x^2} cr} $$ Area of the circle : $$eqalign{
& = pi {r^2} cr
& = left( {frac{{22}}{7} imes frac{{5x}}{2} imes frac{{5x}}{2}}
ight) cr
& = frac{{275{x^2}}}{{14}} cr} $$ Area wasted : $$eqalign{
& = left( {frac{{275{x^2}}}{{14}} - 12{x^2}}
ight) cr
& = frac{{107{x^2}}}{{14}} cr} $$ Required percentage : $$eqalign{
& = left( {frac{{107}}{{14}} imes frac{{14}}{{275}} imes 100}
ight)\% cr
& = 39\% cr} $$
[#80] A rectangular paper, when folded into two congruent parts had a perimeter of 34 cm for each part folded along one set of sides and the same is 38 cm when folded along the other set of sides. What is the area of the paper ?
Correct Answer
(A) 140 cm 2
Explanation
Solution: When folded along breadth, we have : $$eqalign{
& 2left( {frac{l}{2} imes b}
ight) = 34 cr
& or,l + 2b = 34.....(i) cr} $$ When folded along length, we have : $$eqalign{
& 2left( {l imes frac{b}{2}}
ight) = 38 cr
& or,2l + b = 38.....(ii) cr} $$ Solving (i) and (ii), we get : $$l$$ = 14 and b = 10 ∴ Area of the paper : = (14 × 10) cm 2 = 140 cm 2