To determine the maximum height reached by the toy rocket, we need to analyze the given quadratic function that models its height over time:
[tex]\[ f(t) = -18t^2 + 48t \][/tex]
This function represents a parabola opening downwards, as indicated by the negative coefficient of the [tex]\( t^2 \)[/tex] term. The maximum height of the rocket corresponds to the vertex of this parabola.
For a quadratic function of the form [tex]\( f(t) = at^2 + bt + c \)[/tex], the vertex, which gives the maximum (or minimum) value, occurs at:
[tex]\[ t = -\frac{b}{2a} \][/tex]
In our function:
- [tex]\( a = -18 \)[/tex]
- [tex]\( b = 48 \)[/tex]
- [tex]\( c = 0 \)[/tex] (though it's not needed for this specific calculation)
We substitute these values into the vertex formula to find the time [tex]\( t \)[/tex] at which the maximum height occurs:
[tex]\[ t = -\frac{48}{2 \cdot -18} = -\frac{48}{-36} = \frac{48}{36} = \frac{4}{3} \][/tex]
So, the maximum height occurs at [tex]\( t = \frac{4}{3} \)[/tex] seconds (or approximately 1.33 seconds).
Next, we need to find the height at this time by substituting [tex]\( t = \frac{4}{3} \)[/tex] back into the original function [tex]\( f(t) \)[/tex]:
[tex]\[ f\left(\frac{4}{3}\right) = -18 \left(\frac{4}{3}\right)^2 + 48 \left(\frac{4}{3}\right) \][/tex]
Calculate [tex]\( \left(\frac{4}{3}\right)^2 \)[/tex]:
[tex]\[ \left(\frac{4}{3}\right)^2 = \frac{16}{9} \][/tex]
Now substitute back in:
[tex]\[ f\left(\frac{4}{3}\right) = -18 \cdot \frac{16}{9} + 48 \cdot \frac{4}{3} \][/tex]
Simplify the terms:
[tex]\[ f\left(\frac{4}{3}\right) = -18 \cdot \frac{16}{9} + 48 \cdot \frac{4}{3} = -32 + 64 \][/tex]
Finally, combine the results:
[tex]\[ f\left(\frac{4}{3}\right) = -32 + 64 = 32 \][/tex]
Therefore, the maximum height that the rocket reaches is:
[tex]\[ \boxed{32} \text{ feet} \][/tex]
Since none of the answer choices provided match 32 feet exactly, it appears there may have been an error or oversight in creating the answer choices. Nevertheless, the correct maximum height based on our calculation is indeed 32 feet.
