Final answer:
The maximum height of the firework is 64 feet.
Step-by-step explanation:
The equation y = -16x^2 + 32x + 48 represents the height, in feet, of a firework x seconds after it is launched. To find the maximum height of the firework, we need to determine when the vertical velocity (Vy) is zero. This happens when the derivative of the equation with respect to time is zero. So, we differentiate the equation, set it equal to zero, and solve for x. The x value obtained represents the time at which the firework reaches its maximum height. We substitute this x value back into the equation to find the maximum height.
To find the derivative of the equation, we use the power rule for derivatives.
y' = -32x + 32
Setting y' equal to zero and solving for x:
-32x + 32 = 0
-32x = -32
x = 1
Substituting x = 1 into the original equation:
y = -16(1)^2 + 32(1) + 48 = 64
Therefore, the maximum height of the firework is 64 feet.