Final answer:
To find the maximum height of the missile's trajectory given by y = 464x - 16x^2, calculate the vertex of the parabola which occurs at x = 14.5. Substituting this x value into the equation gives a maximum height of y = 233 meters.
Step-by-step explanation:
The student has asked how to find the maximum height of a guided missile's trajectory described by the parabolic equation y = 464x − 16x2. To find this, we need to determine the apex of the parabola, which is where the derivative of the equation with respect to x equals zero (the point where the vertical velocity is zero). This occurs at the vertex of the parabola.
Taking the derivative of the function and setting it to zero:
- dy/dx = 464 - 32x = 0
- x = 464 / 32
- x = 14.5
Substituting this value back into the original equation:
- y = 464(14.5) - 16(14.5)2
- y = 6728 - 3344
- y = 233 m
This means the object's maximum height is 233 meters.