Question

Find the area of the largest rectangle that can be inscribed in a right triangle with legs of lengths 4 cm and 6 cm if two sides of the rectangle lie along the legs. webassign cengage

Answer 1

Answer:

$6cm^2$

Step-by-step explanation:

Let x and y be the sides of the rectangle.

Area of the Triangle, A(x,y)=xy

From the diagram, Triangle ABC is similar to Triangle AKL

AK=4-y

Therefore:

$\dfrac{x}{6} =\dfrac{4-y}{4}$

$4x=6(4-y)\\x=\dfrac{6(4-y)}{4} \\x=1.5(4-y)\\x=6-1.5y$

We substitute x into A(x,y)

$A=y(6-1.5y)=6y-1.5y^2$

We are required to find the maximum area. This is done by finding

the derivative of Aand solving for the critical points.

Derivative of A:

$A'(y)=6-3y\\ Set A'=0\\6-3y=0\\3y=6\\y=2 cm$

Recall that: x=6-1.5y

x=6-1.5(2)

x=6-3

x=3cm

Therefore, the maximum rectangle area is:

Area =3 X 2 =$6cm^2$