Consider the equilateral triangle with vertices at (-1,0),(1,0),(0,sqrt(3)).(This triangle has area sqrt(3).Describe the rectangle with largest area that can be inscribed in the equilateral triangle. What fraction of the triangle's area does this rectangle occupy?
2007-11-13 03:02:05 · 1 answers · asked by Matty B 3 in Science & Mathematics ➔ Mathematics
Let (a,b) be a point on the segment joining A(1,0) and B(0,sqrt(3)), and consider the inscribed rectangle with vertices (a,b), (-a,b), (-a,0), and (a,0). The equation of the line AB is y = -sqrt(3)x + sqrt(3), so b = -sqrt(3)a + sqrt(3).
The area Y of the rectangle is Y = 2ab = 2a(-sqrt(3)a + sqrt(3). The derivative
dY/da = -4sqrt(3)a + 2sqrt(3), and this vanishes when a = 1/2. The area when a = 1/2 is sqrt(3)/2, so the rectangle occupies 1/2 of the triangle's area.
2007-11-13 03:48:53 · answer #1 · answered by Tony 7 · 0⤊ 0⤋