Final answer:
To find the greatest height of a toy rocket modeled by the equation h = -16t² + 32t + 84, we calculate the vertex of the parabola. The maximum height occurs at t = -32/(2*(-16)) = 1 second, resulting in a greatest height of 100 feet (Option 4).
Step-by-step explanation:
The question involves finding the greatest height of a toy rocket, whose height as a function of time is given by a quadratic equation. This scenario is an example of projectile motion, specifically the vertical motion of the rocket launching and ascending to its peak before descending again. The greatest height that the rocket reaches is known as the apex or vertex of the parabolic path it follows.
To find the greatest height of the rocket, we need to determine the vertex of the quadratic function. The given function has the form h = -16t² + 32t + 84, which is a parabola opening downwards because the coefficient of the t² term is negative. The apex of this parabola occurs at the time value t when the derivative of the function with respect to time is zero.
In this case, since the quadratic is in standard form, ax² + bx + c, the t-value of the vertex can also be found using the formula -b/(2a). Plugging the coefficients from the given function into this formula gives us t = -32/(2 × (-16)) = 1 second. Substituting this time back into the height equation gives us the greatest height: h = -16(1)² + 32(1) + 84 = -16 + 32 + 84 = 100 feet.
Therefore, the rocket's greatest height is 100 feet, which corresponds to Option 4.