trigonometry
posted by Britt .
A sheet metal worker constructed a triangular metal flashing with sides of 6 ft., 5 ft., and 3 ft. If he charges $2 per square foot for this type of work, then what should he charge? Do not use Heronâ€™s formula. Justify and explain your reasoning.

Draw a diagram. you have an almostisosceles triangle, with base 3, sides 5,6.
Drop an altitude h to the base side 3. It will divide the base into two sections, of length x and 3x.
Let x be the distance from side 5 to the base of h.
x^2 + h^2 = 25
(3x)^2 + h^2 = 36
25  x^2 = 36  (3x)^2
25  x^2 = 36  9 + 6x  x^2
25 = 27 + 6x
6x = 2
x = 1/3
1/9 + h^2 = 25
h = 4.988
So, the area is 1/2 * 3 * 4.988 = 7.482
That's the area, so you can figure the price.
check: Heron's says a^2 = 7*1*2*4 = 56, so a = 7.483