x²+(x²/(x+1)²) = 3
(x+1)²*x²+x² = 3(x+1)²
x²(x²+2x+1)+x² = 3(x²+2x+1)
x⁴+2x³+2x² = 3x²+6x+3
x⁴+2x³-x²-6x-3 = 0
It implies;
x = ½(1-√(5)) = -0.6180339887
or
x = ½(1+√(5)) = 1.6180339887...
Let the centre of the circle be N
Therefore;
N coordinate is;
N(2, 4).
Point P is;
P(-2, (17/2))
Point Q is;
Q(x, 0)
Gradient of length NP is;
(4-(17/2))/(2+2)
= -9/8...
πa² = 73π
a = √(73) units.
a is the radius of the circle.
b = 2a
b = 2√(73) units.
b is the diameter of the circle.
c²+7² = (2√(73))²
c² = 292-49
c² = 243
c = √(243)
c = 9√(3) units....
Calculating green area.
c² = a²+b²
c = √(4²+6²)
c = √(52)
c = 2√(13) units.
c = 7.2111025509 units.
d = 90-atan(4/6)
d = atan(3/2)°
e = 180-2d
e = (180-2ata(3/2))°
sin(0.5(180-2a...
Calculating AB.
Notice.
1 unit is the side length of each of the composite plane shape.
a = ½*180(6-2)
a = 120°
a is the single interior angle of the regular hexagon.
b = 180(12-2)/12...
Blue Area is;
Area square of single side 4cm - 2(area circle of radius 1cm) - 8(area square of single side 1cm) + 8(area quarter circle of radius 1cm).
= 16 - 2π - 8 + ¼(8π)
= 8 - 2π + 2π
=...
Shaded area is;
Area circle with radius 1 units - 8(area quarter circle with radius 1 units - area triangle with height and base 1 units each).
= π - 8(¼(π)-½)
= π - 2π + 4
= (4-π) square u...
Let a be the radius of the quarter circle.
b = ½(a) units.
tanc = b/a
c = atan(½)°
d = atan(2)°
e = d-c
e = (atan(2)-atan(½))°
Therefore, the required length, x is;
tan(atan(2)-...
Sir Mike Ambrose is the author of the question.
Let the side length of the regular hexagon be 1 unit.
Therefore;
Area green is;
Area triangle with height 1 unit and base ⅓√(3) units.
= ½...
Notice;
Triangle ABF is isosceles.
The height is;
4tan(atan¾)
= 3 units.
The base, given is 8 units.
The two equal side length is;
√(4²+3²)
= √(25)
= 5 units.
Radius, r of...
