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-An explosion causes debris to rise vertically with an initial velocity of 96 feet per second. The function $s ( t ) = - 16 t ^ { 2 } + 96 t$ describes the height of the debris above the ground, $s ( t )$, in feet, $t$ seconds after the explosion. What is the instantaneous speed of the debris $4.8$ second(s) after the explosion?
A) $153.6 \mathrm { ft } / \mathrm { sec }$
B) $- 57.6 \mathrm { ft } / \mathrm { sec }$
C) $57.6 \mathrm { ft } / \mathrm { sec }$
D) $- 153.6 \mathrm { ft } / \mathrm { sec }$

Question

Quizplus · Accepted Answer

To find the instantaneous speed of the debris 4.8 seconds after the explosion, we need to find the derivative of the function $s(t) = -16t^2 + 96t$ with respect to time $t$ and then substitute $t = 4.8$. The derivative, $s'(t) = -32t + 96$, represents the velocity of the debris at any time $t$. Substituting $t = 4.8$ gives $s'(4.8) = -32(4.8) + 96 = -153.6 + 96 = -57.6$ ft/sec, indicating the debris is falling at a speed of 57.6 ft/sec.

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