Question
dajun has family in jamaica who teach him how to play cricket, a popular bat-and-ball sport there. The quadratic function, f (t), can be used to represent the height of a ball in feet, t seconds after dajuan hits the ball. What is the maximum height the ball will reach?
f(t) = -16(t - 3)^3 + 148
type answer here____
f(t) = -16(t - 3)^3 + 148
type answer here____
Answers
GPT-4o mini
To find the maximum height of the ball represented by the function \( f(t) = -16(t - 3)^3 + 148 \), we first notice that this function has the form of a cubic equation rather than a quadratic equation due to the exponent being 3. However, we can still work with it.
The function is in vertex form, but since it is a cubic function, let's clarify that the maximum height can indeed be found. The maximum or minimum of a cubic function occurs either at the inflection point or at critical points where the derivative is zero.
However, since the highest point of a cubic function (when it's leading coefficient is negative) is at the vertex of the turning point of the function, we find the maximum height by evaluating \( f(t) \) at \( t = 3 \).
Calculating \( f(3) \):
\[
f(3) = -16(3 - 3)^3 + 148
\]
\[
f(3) = -16(0) + 148
\]
\[
f(3) = 148
\]
Therefore, the maximum height the ball will reach is:
\[
\boxed{148}
\] feet.
The function is in vertex form, but since it is a cubic function, let's clarify that the maximum height can indeed be found. The maximum or minimum of a cubic function occurs either at the inflection point or at critical points where the derivative is zero.
However, since the highest point of a cubic function (when it's leading coefficient is negative) is at the vertex of the turning point of the function, we find the maximum height by evaluating \( f(t) \) at \( t = 3 \).
Calculating \( f(3) \):
\[
f(3) = -16(3 - 3)^3 + 148
\]
\[
f(3) = -16(0) + 148
\]
\[
f(3) = 148
\]
Therefore, the maximum height the ball will reach is:
\[
\boxed{148}
\] feet.