Question

Flight of a Rocket The altitude in feet attained by a model rocket t seconds into flight is given by the function h(t). Find the maximum altitude (in ft) attained by the rocket. (Round your answer to the nearest foot.) h(t) = − 1 3 t3 + 2t2 + 12t + 21 (t ≥ 0)

Answer 1

Answer:

$h_{max} = 26.114\,ft$

Step-by-step explanation:

The maximum altitude of the rocket can be found with the help of First Derivative and Second Derivative Tests:

$\frac{dh}{dt} = -39\cdot t^{2}+4\cdot t + 12$

$\frac{d^{2}h}{dt^{2}}=-78\cdot t + 4$

Critical values are:

$t_{1}\approx 0.608\,s$, $t_{2}\approx -0.506\,s$

The second derivative values related to critical values are:

$t_{1}:$

$\frac{d^{2}h}{dt^{2}}= -43.424$ (Maximum)

$t_{2}:$

$\frac{d^{2}h}{dt^{2}}= 43.468$ (Minimum)

Only the first solution offers a physically reasonable solution, as time is a positive variable. Then, maximum height is:

$h_{max} = 26.114\,ft$