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The diagonals of a convex quadrilateral are mutually perpendicular. The sum of the lengths of the diagonals is 12. We want to find the maximum possible area of such a quadrilateral. Let us denoted by x and y the lengths of the two diagonals. Then the area of the quadrilateral is the following function of x and y:________.

Respuesta :

Answer:

33

Step-by-step explanation:

Step 1: The Diagonals of the Convex Quadilateral are Mutually Perpendicular this makes the shape a Rhombus Shape

Step 2: The formula for a area of a rhombus is the multiply the two lengths and divide by two.

In this Question we have x×y÷2 or xy/2

Step 3: The sum of the lengths of the diagonals is 12

x+y=12

y=12-x

Therefore, the area, as a function of x alone = x(12-x)/2

Step 4: To find the maximum area, let's find the derivative of the area function in respect to x and then equate to 0.

d/dx(x(12-x)/2)=0

d/dx(12x - x^2) = 0

12-2x=0

12-0=2x

12=2x, divide both sides by 2

x= 6

Step 5: Calculate the Maximum Area

The maximum area will then be 6(12 - 6)/2

= (72-6)/2 = 66/2= 33

So, the maximum area is 33.

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