Question

Among all triangles of hypotenuse 10 units, determines the size of the area is maximized.

Answer 1

I haven't done this type of problem in a long time but I will tryto be helpful

You have to take the derivative of

A = ab/2
h^2 = a^2 + b^2

solve for one of the variables,

h^(2) - b^(2) = a^2
(h^(2) - b^(2))^(1/2) = a

\[ A(b) = \frac{(h^2 - b^2)^{1/2}b}{2}\]

You can take the derivative,

\[\frac{dA}{db} A(b) = \frac{dA}{db} \frac{(h^2 - b^2)^{1/2}b}{2}\]

Find where,
A'(b) = 0
and
A'(b) is undefined but not undefined in the original function

then make a table and see where the function is max and min, the value that gives you the minimum is equal to the value of b that gives the smallest area

Do the same with variable a,

then sub it into Pythagorean theorem and see if it gives you 10

Answer 2

thank you very much :)