From a circular piece of metal sheet with diameter 20cm, a rectangular piece with perimeter 28 cm is to be cut as shown. find the dimensions of the rectangular piece.

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

From a circular piece of metal sheet with diameter 20cm, a rectangular piece with perimeter 28 cm is to be cut as shown. Find the dimensions of the rectangular piece.

Solve the problem of cutting a rectangular piece from a circular metal sheet with a diameter of 20 cm and a perimeter of 28 cm. Learn to apply geometric concepts, including the perimeter and diagonal of a rectangle, and algebraic techniques like solving a quadratic equation. Suitable for students in Grades 9-12.

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