Final answer:
The leftover metal area is r²(π - 2) when expressed in factored form.To calculate the area of the metal pieces left, we first determine the area of the square cut from the circular plate of radius r and then subtract it from the circle's area.
Step-by-step explanation:
A student asked about the area of the metal pieces left after cutting the largest possible square from a circular metal plate with radius r. To find this, we need to understand the relationship between the circle's area and the square's area when the square is inscribed within the circle.
When a square is inscribed in a circle of radius r, the diameter of the circle becomes the diagonal of the square. Using the Pythagorean theorem, the diagonal of the square (which is also the diameter of the circle) is 2r, thus each side of the square is √2r. The area of the square is then (√2r)² = 2r². The remaining area of metal is the area of the circle minus the area of the square.
Therefore, the remaining area is r²π - 2r². To express it in factored form, we factor out r² to get the final answer, which is r²(π - 2).