Find the largest possible real value of
\sqrt{(x-20)(y-x)} + \sqrt{(140-y)(20-x)} + \sqrt{(x-y)(y-140)}
subject to -40\leq x \leq 100 and -20\leq y \leq 200.
80?
432.5
To find the largest possible real value of the given expression, we can use the method of maximizing a function subject to constraints. In this case, the function is the given expression, and the constraints are the given ranges for x and y.
First, let's simplify the given expression by reordering the terms:
\sqrt{(x-20)(y-x)} + \sqrt{(140-y)(20-x)} + \sqrt{(x-y)(y-140)}
Now, let's begin by finding the partial derivatives of the expression with respect to x and y. We want to find the critical points where both partial derivatives are equal to zero.
∂/∂x (\sqrt{(x-20)(y-x)} + \sqrt{(140-y)(20-x)} + \sqrt{(x-y)(y-140)}) = 0
Similarly,
∂/∂y (\sqrt{(x-20)(y-x)} + \sqrt{(140-y)(20-x)} + \sqrt{(x-y)(y-140)}) = 0
Solving these two equations simultaneously will give us the critical points in the range of -40≤x≤100 and -20≤y≤200.
After finding the critical points, we can evaluate the given expression at those points. The largest value of the expression within the given range will correspond to the largest possible real value.
Please note that the solution to this problem involves more complex mathematical calculations. You may use mathematical software or a graphing calculator to solve the system of equations numerically and find the critical points.