Question

the height, h (in feet) of a model rocket launched from the roof of a building at t seconds is given by h = s(t)= - 16(t + 2)(t - 6) s(t)= - 16t² + 64t + 192 b. what is the height of the rocket at t = 0? the initial height is 192 feet c. find the formula for the velocity s(t) s(t)= - 32t + 64 d. what is the initial velocity? s(0)= 64 ft per sec e. the value of a in the formula s(t)=at²+bt + c tells us if the graph opens up or down. it is 1/2 the acceleration of the object due to gravity. what does b represent? a. b is the maximum height b. b is the initial height of the object c. b is the initial velocity and the slope of the tangent line to the function at t = 0 d. b is the velocity when the height is maximum

Explanation:

Step1: Recall position - velocity relationship

The position function of an object is given by $s(t)=at^{2}+bt + c$. The velocity function $s^{\prime}(t)$ is the derivative of the position function. Using the power - rule $\frac{d}{dt}(t^{n})=nt^{n - 1}$, we have $s^{\prime}(t)=2at + b$.

Step2: Evaluate velocity at t = 0

When $t = 0$, $s^{\prime}(0)=b$. The velocity of the object at $t = 0$ is the initial velocity. Also, the derivative of a function at a point gives the slope of the tangent line to the function at that point. So $b$ is the initial velocity and the slope of the tangent line to the function $s(t)$ at $t = 0$.

Answer:

C. b is the initial velocity and the slope of the tangent line to the function at t = 0