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Triangle ABC is made of a piece of elastic string. Vertex A and B are being pulled apart so the length of AB is increasing at a rate of $2$ cm per second and the height, h$,$ is decreasing at a rate of $1$ cm per second. Initially, AB$=30$cm and h $=20$cm$.$

$($a$)$ Find an expression for AB in terms of t and an expression for h in terms of t$,$ respectively.

$($b$)$ Find an expression for the rate of change of the area of the triangle in terms of t$.$

$($c$)$ Find the rate at which the area of the triangle is changing when AB$=34$cm and h $=18$cm$.$