Carpet

The room is 10 x 5 meters. You have the role of carpet width of 1 meter. Make rectangular cut of roll that piece of carpet will be longest possible and it fit into the room. How long is a piece of carpet? Note .: carpet will not be parallel with the diagonal.

Correct result:

x =  10.18034 m

Solution:

$a+b=10 \ \\ c+d = 5 \ \\ x^2 = b^2 +c^2 \ \\ 1^2 = a^2 +d^2 \ \\ \ \\ y = x^2 = (10-a)^2 + (5-\sqrt{ 1-a^2})^2 \ \\ y = x^2 = -10 \sqrt{ 1-a^2} - 20 a + 126 \ \\ \ \\ y' = \dfrac{ 10 a}{ \sqrt(1-a^2)} -20 \ \\ y' = 0 \ \\ a = \dfrac{2}{\sqrt{5}} = 0.894427191 \ m \ \\ x = \sqrt{y} = \sqrt{ -10 \sqrt(1-a^2)-20 a+126 } \ \\ x = 10.18034 \ \text{m}$

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