Could anyone help me with this?

Question

You are given a rectangular piece of paper that has length  cm and height  cm. The lower right corner is to be folded to the top edge forming a triangle as shown. Determine the maximum and minimum area of a triangle that can be constructed.

Answer 1

Setup:

1. Let the lower right corner of the rectangle at (x, 0) = (24.9, 0) be folded to a point (a, y) on the top edge, where 0 ≤ a ≤ x (since the top edge runs from (0, y) to (x, y)).
2. When the corner is folded, the hypotenuse of the triangle becomes the crease of the fold. The triangle formed has one vertex at (24.9, 0), another at the fold's intersection with the top edge (a, y), and the last vertex at the base (a, 0).

Area of the Triangle:

The base of the triangle is the horizontal distance between (24.9 ,0) and (a, 0), which is:

Base = ∣24.9 − a∣

The height of the triangle is the vertical distance between (a, y) and (a, 0), which is:

Height = y = 20 cm

The area of the triangle is:

Area = 1/2 × Base × Height = 1/2 × ∣24.9 − a∣ × 20

Area = 10 × ∣24.9 − a∣ [simplified]

Maximizing and Minimizing the Area:

The function 10 × ∣24.9 − a∣ is maximized when the distance ∣24.9 − a∣ is largest. This occurs at the endpoints a = 0 or a = 24.9, giving:

Maximum Area = 10 × 24.9 = 249 cm²

The function 10 × ∣24.9 − a∣ is minimized when the distance ∣24.9 − a∣ is smallest. This occurs when a = 24.9, resulting in:

Minimum Area = 10 × 0 = 0 cm²

The maximum area of the triangle is 249 cm², and the minimum area is 0 cm².

Hope this was helpful.