Question

Prince is trying to throw a ball over a height of the ball, in feet based on the number of seconds, is represented by the equation F(t) = -3.5t^2 + 14t + 3. What is the maximum height he will throw?

Answer 1

Final answer:

The maximum height
`\( F_\text{max} \)` is 17 feet, attained at time
`\( t_\text{max} = 2 \)` seconds.

Step-by-step explanation:

The equation
`\( F(t) = -3.5t^2 + 14t + 3 \)` represents the height of the ball, in feet, at time
`\( t \)` in seconds. This equation is in the form
`\( F(t) = at^2 + bt + c \)`, where
`\( a = -3.5 \), \( b = 14 \)`, and
`\( c = 3 \)`.

To find the maximum height of the ball, we need to determine the vertex of the parabolic function. The vertex of a parabola in the form
`\( f(t) = at^2 + bt + c \)` is given by the coordinates
`\((t_v, F(t_v))\)`, where
`\( t_v = -(b)/(2a) \)`.

In this case:

`\[ t_v = -(14)/(2(-3.5)) = -(14)/(-7) = 2 \]`

Now, substitute
`\( t = 2 \)` into the original equation to find the maximum height:

`\[ F(2) = -3.5(2)^2 + 14(2) + 3 \]`

Calculate this expression:

`\[ F(2) = -3.5(4) + 28 + 3 = -14 + 28 + 3 = 17 \]`

Therefore, the maximum height the ball will reach is 17 feet.