Question

Find the area of the largest rectangle that can be inscribed in a right triangle with legs of lengths 4 cm and 5 cm if two sides of the rectangle lie along the legs.

Answer 1

Answer:

$A=3 cm^{2}$

Step-by-step explanation:

We need to maximize the area of the rectangle inside the triangle, in this case, it is:

$A=(4-x)y$

the length of this rectangle is (4-x) and width is y.

No, we need to represent y in terms of x. We can use this ratio property here:

$\frac{4}{3}=\frac{x}{y}$

Let's solve it for y:

$y=\frac{3x}{4}$

Then the area will be:

$A=(4-x)\frac{3x}{4}$

$A=3x-\frac{3x^{2}}{4}$

Now, let's take the derivative of A whit respect to x.

$\frac{dA}{dx}=3-\frac{3x}{2}$

And equal to zero to get the maximum value of x and solve it for x.

$0=3-\frac{3x}{2}$

$x=2$

Therefore, the area of largest rectangle will be

$A=3(2)-\frac{3(2)^{2}}{4}$

$A=3 cm^{2}$

I hope it helps you!

Answer 2

We want to find the area of the largest rectangle that can be inscribed on a right triangle of catheti of 4cm and 5cm.

The area of the largest rectangle that we can inscribe in the given right triangle is:

A = 2cm*(5/2)cm = 5cm^2

We know that for a rectangle of length L and width W the area is given by:

A = L*W

Now let's say that L is along the 5cm side and W is along the 4cm side, let's find a relation between these two.

The quotient between the catheti of the right triangle must be equal to the quotient of the catheti of the right triangle that is generated when we inscribe the rectangle in the larger triangle.

That new triangle will have legs equal to:

5cm - L and W.

Then we have that:

$\frac{5cm - L}{W} = \frac{5cm}{4cm} \\\\L = 5cm - \frac{5}{4}*W$

Now we can write L in terms of W, then we can replace that in the area equation to get

A = (5cm - (5/4)*W)*W = 5cm*W - (5/4)*W^2

This is the equation we want to maximize, notice that this is a quadratic polynomial of negative leading coefficient, meaning that the maximum is at the vertex.

And as you may know, the vertex of the general quadratic polynomial:

a*x^2 + b*x + c

is at:

x = -b/2a

Then in this case, the vertex is at:

W = (-5cm)/(2*5/4) = 2cm

Then the length is given by:

L =  5cm - (5/4)*2cm = 5cm - (5/2)cm = (5/2)cm

Then the area of the largest rectangle that we can inscribe in the given right triangle is:

A = 2cm*(5/2)cm = 5cm^2

If you want to learn more, you can read:

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