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Mathematics Question and Solution

https://onlineedumath.com/update/mathematics-question-and-solution/1999278

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Considering similar triangles ADC and CDB ratios. Let CD be a. 16 = a   a = 4 Cross Multiply. a² = 64 a = 8 units c² = 16²+8² c = 8√(5) units. c = AC. Calculating r1. 16r1+8r1+8√(5)r1 = (16*8) r1(24+8√(5)) = 128 r1 = 128/(24+8√(5)) r1 = 16/(3+√(5)) r1 = (12-4√(5)) units. d² = 8²+4² d = 4√(5) units. d is BC. Calculating r2. 4r2+8r2+4√(5)r2 = (8*4) r2(12+4√(5)) = 32 r2(3+√(5)) = 8 r2 = 8/(3+√(5)) r2 = (6-2√(5)) units. Calculating EF. (EF)² = (r1+r2)²+(r1-r2)² (EF)² = ((12-4√(5))+(6-2√...

Mathematics Question and Solution

https://onlineedumath.com/update/mathematics-question-and-solution/1997920

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Calculating a. (1/sin120) = (a/sin20) a = 0.39493084363 units. Calculating b. (1/sin120) = (b/sin40) b = 0.74222719897 units. Therefore, observing the proof; (1/a)+(1/b) = 3ab+3 (1/a)+(1/b) is; 1/0.39493084363+1/0.74222719897 = 3.8793852416   3ab+3 is; 3*0.39493084363*0.74222719897+3 = 3.87938524156  Proved.

Mathematics Question and Solution

https://onlineedumath.com/update/mathematics-question-and-solution/1997583

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Notice; a² = 5² - (⅕(√(481)))² a = ⅕(12) units. b = 5-a b = ⅕(13) units. c² = 4²-(⅕(12))² c = ⅕(16) units. d = atan(4/3)° (a) Area A exactly is; ½*⅕(16)*(5+⅕(13)) - atan(4/3)π*16/360 = ((8*38)/25) - (2atan(4/3)π/45) = (304/25) - ((2atan(4/3)π)/45) = (2736-10πatan(4/3))/225 square units. = 4.74163825599 square units. (b) Calculating Area B to 2 decimal places. ½*10(10+5) - ½*⅕(16)(5+⅕(13)) - ((atan½+ata¾)π*16/360) - (⅕(13)*⅕(34-√(481))) - ½*⅕(√(481))(5+⅕(13)) - ((atan(12/√(481))+(2t...

Mathematics Question and Solution

https://onlineedumath.com/update/mathematics-question-and-solution/1997564

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Let AB be 1 unit. BC = 3 units. It implies; AC = 1+3 AC = 4 units. AC is a side length of the square. Radius of the inscribed half circle a is; ½(BC) a = ½(3) units. a = 1.5 units. b = 1.5+AB b = 1.5+1 b = 2.5 units. b = ½(5) units. c²+(1.5)² = (2.5)² c² = 2.5²-1.5² c = 2 units. Considering similar triangle ratios. 2 = 4 1.5 = d Cross Multiply. d = 3 units. tane = (3/4) e = atan(3/4)° f = atan(4/3)° Calculating g, side length of the regular triangle. (4/sin120) = (g/sin(atan(4/3)))...

Mathematics Question and Solution

https://onlineedumath.com/update/mathematics-question-and-solution/1996975

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Let a be 1 unit. (1/sin48) = (b/sin24)  b = 0.54731813925 units. (1/sin48) = (c/sin108) c = 1.27977277603 units. DC = 1.27977277603+0.54731813925+1 DC = 2.82709091528 units. Therefore, the required angle x is; (2.82709091528/sin36) = (1/sinx) x = 12°

Mathematics Question and Solution

https://onlineedumath.com/update/mathematics-question-and-solution/1996244

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Let the radius R, of the half circle be 1 unit. sin30 = c/1 c = ½ units. c = 0.5 units. cos30 = d/1 d = ½√(3) units. d = 0.86602540378 units. e = 1-0.86602540378  e = 0.13397459622 units. f² = (0.13397459622)²+(0.5)² f = 0.51763809021 units. Calculating r, radius of the small inscribed yellow circle. Considering similar triangle ratios. 0.51763809021 = (1-r) 0.5 = (0.5+r) Cross Multiply. 0.5-0.5r = 0.51763809021r+0.2588190451  0.5-0.2588190451 = 0.5r+0.51763809...

Mathematics Question and Solution

https://onlineedumath.com/update/mathematics-question-and-solution/1995689

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Let the small inscribed circle's radius be x. x = 6 cm. Let the big inscribed circle's radius be y. y = 8 cm. Let the ascribed semi circle's radius be z.  = 18 cm. a² = 10²-8² a = 6 cm. b² = 12²-6² b = 6√(3) cm c = atan(z/b)° c = atan(18/6√(3))° c = atan√(3)° c = 60° d = atan(z/a)° d = atan(18/6) d = atan(3)° (a) Angle theta exactly is; Let it be e. e = 180-c-d e = 180-60-atan3 e = (120-atan3)° e = (atan(⅓)+30)° (b) Calculating Area Shaded to 1 decimal place cm². f = atan(1/√(3)) = 3...

Mathematics Question and Solution

https://onlineedumath.com/update/mathematics-question-and-solution/1994045

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Let r be the radius of the circle. 3x+3 = 2r --- (1). r² = (r-1)²+(r-2x)² r² = r²-2r+1+r²-4xr+4x² 4xr+2r = r²+4x²+1 --- (2). From (1). 3x+3 = 2r x = ⅓(2r-3) --- (3). Calculating r. Therefore, Substituting (3) in (2). 4( ⅓(2r-3))r+2r = r²+4(⅓(2r-3))²+1 ⅓(8r²-6r) = r²+4(⅑(4r²-12r+9))+1 ⅓(8r²-6r) = ⅑(9r²+16r²-48r+36+9) ⅓(8r²-6r) = ⅑(25r²-48r+45) 24r²-18r = 25r²-48r+45 r²-30r+45 = 0 Solving the quadratic equation. r ≠ (15+6√(5)) units. r = (15-6√(5)) units. r = 1.583592135 units.

Mathematics Question and Solution

https://onlineedumath.com/update/mathematics-question-and-solution/1993663

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Notice; Hexagon Area = 24√(3) square units. Calculating hexagon side length. Let it be a. (6a²)/(4tan30) = 24√(3) ½(3√(3))a² = 24√(3) 3a² = 48 a = 4 units. Equation of the curve is; y = ½(x²) ------ (1). sin60 = b/4 b = 2√(3) units. cos60 = c/4 c = 2 units. Gradient = ((4√(3)+2)-(2√(3)+2))/(2-4) Gradient = -√(3) (y-(4√(3)+2))/(x-2) = -√(3) y = -√(3)x+6√(3)+2 ---- (2). Equation (1) and (2). ½(x)² = -√(3)x+6√(3)+2 x = 3.53906 y = ½(3.53906)² y = 6.2624728418 d = 2(3.53906) d = 7....

Mathematics Question and Solution

https://onlineedumath.com/update/mathematics-question-and-solution/1993189

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Let b = 1 unit. (1/sin72) = (a/sin48) a = 0.78138871113 units. BD = 1-0.78138871113  BD = 0.21861128887 units. (AD)² = 1+0.21861128887²-2*0.21861128887cos60 AD = 0.91059299731 units. Calculating the required angle x. (0.91059299731/sin60) = (0.21861128887/sinx) x = 12°
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