Answer:
The reasonable domain is [tex]0\leq x\leq 30[/tex].
Step-by-step explanation:
The given function is
[tex]f(x)=-16x^2+14400[/tex]
Where, f(x) is height (in feet) of the food packet and x is time( in seconds).
Since the leading coefficient is negative, therefore it is an downward parabola.
The time is always positive value, therefore the domain of f(x) must be greater than or equal to zero.
[tex]0\let x[/tex] .... (1)
The given equation can be written as
[tex]f(x)=-16(x^2-900)[/tex]
[tex]f(x)=-16(x^2-(30)^2)[/tex]
[tex]f(x)=-16(x-30)(x+30)[/tex]
Equate each factor equal to 0, to find the x-intercepts.
[tex]x=30,-30[/tex]
From the graph of f(x) is noticed that the x-intercepts are -30 and 30. Before -30 and after 30 the graph of f(x) gives negative values. Therefore the domain must be lies between -30 to 30.
[tex]-30\let x\leq 30[/tex] .... (2)
From (1) and (2), we get
[tex]0\leq x\leq 30[/tex]
Therefore the reasonable domain is [tex]0\leq x\leq 30[/tex].
