To determine the maximum height that the toy rocket reaches, we need to analyze the function that models its height over time: [tex]\( f(t) = -16t^2 + 48t \)[/tex].
This is a quadratic function of the form [tex]\( f(t) = at^2 + bt + c \)[/tex], where [tex]\( a = -16 \)[/tex], [tex]\( b = 48 \)[/tex], and [tex]\( c = 0 \)[/tex]. Since the coefficient of [tex]\( t^2 \)[/tex] (which is [tex]\( a \)[/tex]) is negative, this parabola opens downwards, meaning it has a maximum point.
The vertex of a parabolic function [tex]\( f(t) = at^2 + bt + c \)[/tex] is given by the formula:
[tex]\[ t = -\frac{b}{2a} \][/tex]
Let's find the time [tex]\( t \)[/tex] when the rocket reaches its maximum height by substituting [tex]\( a = -16 \)[/tex] and [tex]\( b = 48 \)[/tex] into the formula:
[tex]\[ t = -\frac{48}{2 \cdot (-16)} \][/tex]
[tex]\[ t = -\frac{48}{-32} \][/tex]
[tex]\[ t = 1.5 \][/tex]
Thus, the rocket reaches its maximum height at [tex]\( t = 1.5 \)[/tex] seconds.
To find the maximum height, we substitute [tex]\( t = 1.5 \)[/tex] back into the height function [tex]\( f(t) \)[/tex]:
[tex]\[ f(1.5) = -16(1.5)^2 + 48(1.5) \][/tex]
First, calculate [tex]\( 1.5^2 \)[/tex]:
[tex]\[ 1.5^2 = 2.25 \][/tex]
Then, substitute and evaluate:
[tex]\[ f(1.5) = -16 \cdot 2.25 + 48 \cdot 1.5 \][/tex]
[tex]\[ f(1.5) = -36 + 72 \][/tex]
[tex]\[ f(1.5) = 36 \][/tex]
Therefore, the maximum height that the toy rocket reaches is 36 feet.
So, the correct answer is:
[tex]\[ \boxed{36 \text{ ft}} \][/tex]
