Answer:
The maximum height reached by the projectile is 80 feet
Step-by-step explanation:
The given equation of the path of the projectile is y = -16·x² + 64·x + 16
Where;
y = The height in feet, reached by the projectile (Assumption)
x = The time it takes the projectile to reach the height, y (Assumption)
The shape of the given equation of the path of the parabola is that of a parabola turned upside down.
We have that the maximum height is given by the top of the curve where the curve changes direction, and the slope = 0
Therefore, we have;
[tex]Slope = \dfrac{\mathrm{d} y}{\mathrm{d} x} = \dfrac{\mathrm{d} \left (-16\cdot x^2 + 64\cdot x + 16 \right )}{\mathrm{d} x} = -32 \cdot x + 64 = 0[/tex]
Therefore, at the maximum point, the slope is -32·x + 64 = 0
∴ x = -64/(-32) = 2 at the maximum point
The height at the maximum point = The maximum height, [tex]y_{max}[/tex], is found by finding the value of y (the height) at x = 2 (the value of x at the maximum point)
Therefore, we have;
[tex]y_{max}[/tex] = -16 × 2² + 64 × 2 + 16 = 80 feet
The maximum height reached by the projectile, [tex]y_{max}[/tex] = 80 feet.
