Application of maximization suppose a right triangle with hypotenuse of 12 meters has sides a and b label each sidedetermine a formula that represents the area of this triangle using variables a and brewrite the formula using only one variable. this can be done by first determining a relationship between a and b and then making a substitution domain and range

Since this is a right triangle, 

a^2 + b^2 = 12^2 = 144, so b^2 = 144 - a^2, and  b = sqrt(144-a^2)
        
                                                  ab      a*sqrt(144-a^2)
The area of the triangle is A = =
                                                   2                 2

Since the radicand of the square root function cannot be smaller than zero, the domain for a is [0, 12]; the domain for b is also [0,12].  If a increases, b must decrease, and vice versa.

The area A of the triangle can never be less than 0 nor more than what?
You could use calculus to answer this question, or you could make a table of a, b and Area values and then determine from the table which area A is the max.