Mathematics Question and Solution
Calculating the width of the ascribed rectangle.
a = 180-60-x
a = (120-x)°
Let b be the side length of the inscribed regular triangle.
cosx = 1/b --- (1).
At (1).
1 is adjacent.
b is hypotenuse.
Calculating c, opposite.
c²+1² = b²
c = √(b²-1) units.
c is the width of the rectangle.
Therefore;
sinx = √(b²-1)/b --- (2).
cos(120-x) = 2/b --- (3).
At (3).
cos120cosx+sin120sinx = 2/b
½√(3)sinx-½(cosx) = 2/b --- (4).
Substituting (2) and (1) in (4).
½√(3)(√(b²-1)/b)-½(1/b) = 2/b
√(3)(√(b²-1))-1 = 4
√(3)(√(b²-1)) = 5
b²-1 = (5/√(3))²
b² = ⅓(25)+1
b² = ⅓(28)
b = √(⅓(28))
b = ⅓(2√(21)) units.
b = 3.0550504633 units.
Again, b is the side of the inscribed regular triangle.
Therefore, the required length, the width of the rectangle c is;
c = √(b²-1)
And b = ⅓(2√(21)) units.
c = √((⅓(2√(21)))²-1)
c = ⅓(5√(3)) units.
c = 2.88675134595 units.
Again, c is the required width of the rectangle.
