Gọi độ dài đáy bé của hình thang là x cm.
Vậy độ dài đáy lớn của hình thang là (42 - x) cm.
Theo đề bài, khi mở rộng đáy bé thêm 5 cm, diện tích hình thang tăng thêm 30 cm².
Ta có công thức tính diện tích hình thang: S = (đáy lớn + đáy bé) * chiều cao / 2.
Vậy diện tích hình thang ban đầu là: S1 = ((42 - x) + x) * h / 2 = 21h.
Diện tích hình thang sau khi mở rộng đáy bé là: S2 = ((42 - x + 5) + x) * h / 2 = (47 + x) * h / 2.
Theo đề bài, S2 - S1 = 30.
Vậy (47 + x) * h / 2 - 21h = 30.
Đặt t = h / 2, ta có (47 + x) * t - 21t = 30.
Simplifying the equation, we get 47t + xt - 21t = 30.
Combining like terms, we get 26t + xt = 30.
Rearranging the equation, we get xt = 30 - 26t.
Dividing both sides by x, we get t = (30 - 26t) / x.
Multiplying both sides by x, we get xt = 30x - 26t.
Combining like terms, we get xt + 26t = 30x.
Factoring out t, we get t(x + 26) = 30x.
Dividing both sides by (x + 26), we get t = 30x / (x + 26).
Substituting t = h / 2, we get h / 2 = 30x / (x + 26).
Cross multiplying, we get (x + 26)(h / 2) = 30x.
Expanding the equation, we get (xh + 26h) / 2 = 30x.
Multiplying both sides by 2, we get xh + 26h = 60x.
Rearranging the equation, we get xh - 60x = -26h.
Factoring out x, we get x(h - 60) = -26h.
Dividing both sides by (h - 60), we get x = -26h / (h - 60).
Since the length cannot be negative, we can conclude that h - 60 ≠ 0.
Therefore, x = -26h / (h - 60).
To find the area of the trapezoid, we can use the formula A = (a + b) * h / 2, where a and b are the lengths of the bases and h is the height.
Substituting the given values, we get A = (42 + x) * h / 2.
Substituting x = -26h / (h - 60), we get A = (42 - 26h / (h - 60)) * h / 2.
Multiplying both sides by 2, we get 2A = (42 - 26h / (h - 60)) * h.
Expanding the equation, we get 2A = (42h - 26h^2 / (h - 60)).
Multiplying both sides by (h - 60), we get 2A(h - 60) = (42h - 26h^2).
Expanding the equation, we get 2Ah - 120A = 42h - 26h^2.
Rearranging the equation, we get 26h^2 - 42h + 2Ah - 120A = 0.
Since we want to find the area of the trapezoid, we can assume that A ≠ 0.
Therefore, we can divide both sides of the equation by A, we get 26h^2 / A - 42h / A + 2h - 120 = 0.
Factoring out h, we get h(26h / A - 42 / A + 2) - 120 = 0.
Combining like terms, we get h(26h / A - 42 / A + 2) = 120.
Dividing both sides by (26h / A - 42 / A + 2), we get h = 120 / (26h / A - 42 / A + 2).
Therefore, the area of the trapezoid is A = (42 - x) * h / 2 = (42 - x) * 120 / (26h / A - 42 / A + 2).