We are tasked with cutting a rectangular piece from a circular metal sheet with a diameter of 20 cm, such that the perimeter of the rectangle is 28 cm. The rectangle is inscribed inside the circle.

Let's work through the steps:

Step 1: Key Information

• The diameter of the circular sheet is 20 cm.
• The perimeter of the rectangle is 28 cm.

Step 2: Perimeter of the Rectangle

The formula for the perimeter of a rectangle is given by:

$P = 2 \times (l + w)$ where $l$ is the length, and $w$ is the width of the rectangle. Since the perimeter is 28 cm, we have:

$2 \times (l + w) = 28$

Dividing both sides by 2:

$l + w = 14$ This is the first equation relating $l$ and $w$.

Step 3: Relationship from the Circle

The rectangular piece is inscribed in the circle. The diagonal of the rectangle will be the diameter of the circle. The formula for the diagonal $d$ of a rectangle is:

$d = \sqrt{l^2 + w^2}$

Since the diagonal is equal to the diameter of the circle, which is 20 cm, we get:

$\sqrt{l^2 + w^2} = 20$

Squaring both sides:

$l^2 + w^2 = 400$ This is the second equation.

Step 4: Solve the System of Equations

We now have the system of equations:

1. $l + w = 14$
2. $l^2 + w^2 = 400$

To solve this, let's express $w$ in terms of $l$ from the first equation:

$w = 14 - l$

Substitute this into the second equation:

$l^2 + (14 - l)^2 = 400$

Expanding the square:

$l^2 + (196 - 28l + l^2) = 400$

Combine like terms:

$2l^2 - 28l + 196 = 400$

Simplifying:

$2l^2 - 28l - 204 = 0$

Dividing the entire equation by 2:

$l^2 - 14l - 102 = 0$

Step 5: Solve the Quadratic Equation

We can now solve the quadratic equation $l^2 - 14l - 102 = 0$ using the quadratic formula:

$l = \frac{-(-14) \pm \sqrt{(-14)^2 - 4(1)(-102)}}{2(1)}$

Simplifying:

$l = \frac{14 \pm \sqrt{196 + 408}}{2}$ $l = \frac{14 \pm \sqrt{604}}{2}$ $l = \frac{14 \pm 24.57}{2}$

So, we have two possible solutions for $l$:

1. $l = \frac{14 + 24.57}{2} = 19.285$
2. $l = \frac{14 - 24.57}{2} = -5.285$ (which is not valid as length cannot be negative)

Thus, $l = 19.285$ cm.

Step 6: Calculate $w$

Using $l + w = 14$:

$19.285 + w = 14$ $w = 14 - 19.285 = -5.285 \, \text{cm} \, (which is not feasible for width)$

We need to adjust for dimensional constraint