Mathematics Question and Solution

https://onlineedumath.com/update/mathematics-question-and-solution/3150384

Sir Mike Ambrose is the author of the question. Let the single side length of the square be 2 units. Calculating area S. It will be; Area trapezoid of two parallel side lengths 2 units and (2-√(3)) units and height 1 unit - area sector of radius 2 units and angle 60° + area triangle of height √(3) units and base 2 units. Notice; √(2²-1²) = √(3) units. Area S will be; ½(4-√(3))-(4π)/6+½(2√(3)) = 2-½(√(3))-⅔(π)+√(3) = (12+3√(3)-4π)/6 square units. Calculating area R...

Mathematics Question and Solution

https://onlineedumath.com/update/mathematics-question-and-solution/3149453

Using similar triangle approach. It implies; n ~ 1 n+1 ~ n Cross multiply. Therefore; n²=n+1 n²-n-1=0 (n-½)²=1+¼ n =½±½(√(5)) Therefore; n ≠ ½(1-√(5)) n = ½(1+√(5)) units. Where n is Phi (golden ratio).

Mathematics Question and Solution

https://onlineedumath.com/update/mathematics-question-and-solution/3149449

Let length AB be x. A careful analysis carried out on the given composite plane shape lead to the derivation of right-angled tiangle EQX with hypotenuse QX ⅓(4x) and the two adjacent side of the triangle, EQ ⅔(√(3))x and EX (x-1) respectively. Therefore applying Pythagoras rule to get x (length AB). (⅔(√(3))x)² + (x-1)² = (⅓(4x))² (12x²)/9 + x² - 2x + 1 = (16x²)/9 21x² - 18x + 9 = 16x² 5x² - 18x + 9 = 0 (5x²-15x)-(3x+9) = 0 5x(x-3)-3(x-3)=0 (5x-3)(x-3)=0 Therefore; x ≠ 0.6 x = 3 L...

Mathematics Question and Solution

https://onlineedumath.com/update/mathematics-question-and-solution/3148248

Calculating the radius of the two congruent inscribed circles. Let the radius of the two congruent inscribed circles be a each. b² = 3²+(2+2)² b = √(25) b = 5 units. It implies; (4*(3a))+(3*2)+(5*a) = (3*4) 12a+6+5a = 12 17a = 6 a = 6/17 units.

Mathematics Question and Solution

https://onlineedumath.com/update/mathematics-question-and-solution/3148246

Calculating x, the side length of the ascribed square. a²+x² = 8² a = √(64-x²) units. b²+x² = 10² b = √(100-x²) units. It implies;. c = (x-√(64-x²)) units. d = (x-√(100-x²)) units. Therefore; ⅛(√(64-x²)) = ⅙(x-√(100-x²)) 4(x-√(100-x²)) = 3√(64-x²) 4x-4√(100-x²) = 3√(64-x²) 16x²-32x√(100-x²)+1600-16x² = 576-9x² 9x²+1024 = 32x√(100-x²) 81x⁴+18432x²+1048576 = 102400x²-1024x⁴ 1105x⁴-83968x²+1048576 = 0 Therefore; x² = 60.2353 Or x² = 15.7538 It implies; x ≠ √(60.2353) units. x = √...

Mathematics Question and Solution

https://onlineedumath.com/update/mathematics-question-and-solution/3146954

Let the radius of the quarter circle be 1 unit. b = (a+1) units. a is AD = DE. a² = 2-2cosx 2cosx = 2-a² cosx = ½(2-a²) --- (1). cosx = 1/(1+a) --- (2). Calculating a. Equating (1) and (2). ½(2-a²) = 1/(1+a) 2 = (2-a²)(1+a) 2 = 2+2a-a²-a³ 0 = 2a-a²-a³ a²+a-2 = 0 (a+½)² = 2+(+½)² (a+½)² = ¼(9) a = -½±√(9/4) a = -½±(3/2) Therefore; a ≠ -½-(3/2) a = -½+(3/2) a = 1 unit. Recall. cosx = 1/(1+a) at (1). And a = 1 unit. cosx = 1/(1+1) cosx = ½ x = 60° It...

Mathematics Question and Solution

https://onlineedumath.com/update/mathematics-question-and-solution/3146951

Calculating Area Inscribed Red Circle. a = ½(8) a = 4 units. a is the radius of the two congruent half circles. b = (4+x) units. x is the radius of the red inscribed circle. c = (4-x) units. Therefore; (4-x)² = x²+d² 16-8x+x²-x² = d² d = √(16-8x) units. e = 4+d e = (4+√(16-8x)) units. Calculating x. (4+x)² = (4+√(16-8x))²+x² 16+8x+x² = 16+8√(16-8x)+16-8x+x² 16x = 16+8√(16-8x) 16x-16 = 8√(16-8x) 2x-2 = √(16-8x) 4x²-8x+4 = 16-8x 4x² = 12 x² = 3 x = √(3) units. Again, x is the ra...

Mathematics Question and Solution

https://onlineedumath.com/update/mathematics-question-and-solution/3145977

Calculating r, radius of the circle. Notice that the plane shape is not drawn to scale. r² = (7+a)²+(0.5*10)² r² = 49+14a+a²+25 r² = 74+14a+a² --- (1). Again. r² = a²+(7+5)² r² = a²+144 --- (2). Therefore, equating (1) and (2) to get a. 74+14a+a² = a²+144 14a = 70 2a = 10 a = 5 units. Therefore, r is; Considering (2). r² = a²+144 And a = 5 units. r² = 5²+144 r² = 25+144 r² = 169 r = √(169) r = 13 units.

Mathematics Question and Solution

https://onlineedumath.com/update/mathematics-question-and-solution/3145973

Calculating r, radius of the half circle. Let the two equal lengths be a each. Therefore; r = 2a units. Calculating a. (2a)² = a²+9²-2a*9cosb 3a² = 81+18acosb cosb = (3a²-81)/(18a) cosb = (a²-27)/(6a) --- (1). cosb = 3/a --- (2). Equating (1) and (2). (a²-27)/(6a) = 3/a a²-27 = 18 a² = 45 a = √(45) a = 3√(5) units. Therefore r, radius of the half circle is ; Recall. r = 2a And a = 3√(5) units. r = 2(3√(5)) r = 6√(5) units. r = 13.416407865 units.

Mathematics Question and Solution

https://onlineedumath.com/update/mathematics-question-and-solution/3145292

Calculating shaded area. tana = 5/10 a = atan(½)° Therefore; b = atan(2)° c = 90-2a c = 90-2atan(½) c = 36.8698976458° d = 180-2b d = 180-2atan(2) d = 53.1301023542° Shaded Area is; Area square with length 10 cm - 2(area triangle with height 10 cm and base 5 cm) - Area sector with radius 10 cm and angle 36.8698976458° - Area sector with radius 5 cm and angle 53.1301023542°. = 10²-2(½*10*5)-(36.8698976458π*10²/360)-(53.1301023542π*5²/360) = 100-50-32.1750554396-...

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