Answer:
• Domain: (0,30)
,
• Range: (0, 900)
Explanation:
The equation that models the height of the rocket is:
[tex]h(t)=-4t^2+120t[/tex]
Domain
The domain of the function is the set of all the possible values of t (in seconds) for which the height is defined.
First, find the zeros of the function:
[tex]\begin{gathered} -4t(t-30)=0 \\ -4t=0\text{ or }t-30=0 \\ t=0\text{ or }t=30 \end{gathered}[/tex]
Since the height of the rocket cannot be negative, the domain of h(t) is:
[tex](0,30)[/tex]
Range
The leading coefficient is negative, so the equation has an upside-down parabola.
First, we find the maximum point of the parabola by using the vertex formula.
[tex]Vertex=(-\frac{b}{2a},\frac{4ac-b^2}{4a})[/tex]
From the equation: a=-4, b=120 and c=0
Since we need just the maximum value, we calculate the y-coordinate only:
[tex]\frac{4ac-b^2}{4a}=\frac{4(-4)(0)-(120)^2}{4(-4)}=-\frac{14400}{-16}=900[/tex]
The maximum height of the rocket is 900 ft, therefore, the range of h(t) is:
[tex](0,900)[/tex]
