Mathematics Question and Solution
Calculating R, radius of the circle.
R² = 8²+5²-2*8*5cosx
R² = 89-80cosx --- (1).
a = (90-x)°
R² = 5²+12²-2*5*12cos(90-x)
R² = 169-120cos(90-x)
R² = 169-120(cos90cosx-sin90sinx)
R² = 169-120sinx --- (2).
At (1).
R² = 89-80cosx
80cosx = 89-R²
cosx = (89-R²)/80 --- (3).
At (3).
(89-R²) is adjacent.
80 is hypotenuse.
Calculating opposite.
Opposite = √(80²-(89-R²)²)
Opposite = √(6400-7921+178R²-R⁴)
Opposite = √(-1521+178R²-R⁴) units.
Therefore;
sinx = √(-1521+178R²-R⁴)/80 --- (4).
Substituting (4) in (2).
R² = 169-120(√(-1521+178R²-R⁴)/80)
R² = 169-3(√(-1521+178R²-R⁴)/2)
2R² = 338-3√(-1521+178R²-R⁴)
3√(-1521+178R²-R⁴) = 338-2R²
9(-1521+178R²-R⁴) = (338-2R²)²
-13689+1602R²-9R⁴ = 114244-1352R²+4R⁴
13R⁴-2954R²+127933 = 0
It implies;
R² = 58.2308
Or
R² = 169
Therefore;
R ≠ √(169) ≠ 13 units.
R = √(58.2308) units.
R = 7.63091082375 units.
R is the radius of the circle.
