Question

For ΔABC with three sides a, b, and c, if a=4 and b=3, then the value of c which yields max area is...

Answer 1

let the angle between sides a and b be x
area = (1/2)(a)(b)sinx
area = (1/2)(4)(3)sinx
area = 6sinx
d(area)/dx = 6cosx
for a max area, 6cosx = 0
cosx = 0
x = 90°
so we have a right-angled triangle with c the hypotenuse.
I recognize the 3-4-5 right-angled triangle and c = 5
or
c^2 = 3^2 + 4^2
c^2 = 25
c = √25 = 5