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 Heron’s formula. Justify and explain your reasoning.
Draw a diagram. you have an almost-isosceles 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 3-x.
Let x be the distance from side 5 to the base of h.
x^2 + h^2 = 25
(3-x)^2 + h^2 = 36
25 - x^2 = 36 - (3-x)^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
To calculate the area of the triangular metal flashing, we can use Heron's formula, which states that the area (A) of a triangle with side lengths a, b, and c is given by:
A = √[s(s-a)(s-b)(s-c)]
where s is the semi-perimeter of the triangle, defined as:
s = (a + b + c) / 2
In this case, the side lengths of the triangular metal flashing are 6 ft., 5 ft., and 3 ft. Therefore, we can calculate the semi-perimeter as:
s = (6 + 5 + 3) / 2 = 14 / 2 = 7 ft.
Now, we can substitute the values into the area formula:
A = √[7(7-6)(7-5)(7-3)]
= √[7(1)(2)(4)]
= √[56]
= 7.4833 ft² (rounded to four decimal places)
Since the worker charges $2 per square foot, the total charge for the triangular metal flashing can be calculated by multiplying the area by the cost per square foot:
Total charge = Area × Cost per square foot
= 7.4833 ft² × $2/ft²
= $14.97
Therefore, the sheet metal worker should charge $14.97 for this type of work.