From a circular piece of metal sheet with diameter 20 cm, a rectangular piece with perimeter 28 cm is to be cut as shown. Find the dimensions of the rectangular piece.
"as shown"?
well, a 12x16 rectangle has a diagonal of 20, but its perimeter is 2*28...
To find the dimensions of the rectangular piece, let's first try to visualize the diagram.
From the circular piece of metal sheet, we need to cut a rectangular piece with perimeter 28 cm. We can assume that the rectangle will be positioned inside the circular piece, as shown in the diagram below:
```
-- ======================= --
/ \
| |
| |
| |
| |
\ /
-- ======================= --
```
Let's denote the length of the rectangular piece as "l" and the width as "w." We can see that the length of the rectangular piece will be equal to the circumference of the circular piece, which is 2πr, and the width will be equal to the radius of the circular piece.
Given that the diameter of the circular piece is 20 cm, we can calculate the radius (r) using the formula:
r = diameter / 2
= 20 cm / 2
= 10 cm
Now, let's calculate the length (l) of the rectangular piece:
l = 2πr
= 2π(10 cm)
= 20π cm
Next, let's calculate the width (w) of the rectangular piece, which is equal to the radius (r) of the circular piece:
w = r
= 10 cm
Therefore, the dimensions of the rectangular piece are:
Length (l) = 20π cm (approximately 62.83 cm)
Width (w) = 10 cm
Thus, the rectangular piece has dimensions of approximately 62.83 cm by 10 cm.
To find the dimensions of the rectangular piece, we need to use the information given about the circular piece and the rectangular piece.
Let's denote the dimensions of the rectangular piece as length (L) and width (W). We know that the perimeter of a rectangle is given by the formula 2(L + W).
In this case, the perimeter of the rectangular piece is given as 28 cm. So we can write the equation:
2(L + W) = 28
From the diagram, we can see that the length of the rectangular piece is equivalent to the circumference of the circular piece, which is given by the formula 2πr.
The diameter of the circular piece is given as 20 cm, so the radius (r) is half of the diameter, which is 10 cm. So we have:
L = 2πr = 2π(10) = 20π cm
Now we can substitute the value of L in the equation for the perimeter:
2(20π + W) = 28
Now let's solve this equation to find the width (W) of the rectangular piece:
40π + 2W = 28
2W = 28 - 40π
W = (28 - 40π)/2
W ≈ 28 - 62.83
W ≈ - 34.83
Since width cannot be a negative value, in this case, there is no valid solution for the dimensions of the rectangular piece.