Question

Using the equation representing the height of the firework (h = -16t2 + v0t + h0), algebraically determine the extreme value of f(t) by completing the square and finding the vertex. Interpret what the value represents in this situation.

Answer 1

-16t^2 + v0t + h0

= -16(t^2 - 1/16 v0t) + h0
= -16 [(t - 1/32v0)^2 - (1/32 v0)^2 ) + h0
= -16(t - 1/32v0)^2 + 1/64 (v0)^2 + h0 which is the vertex form

The vertex is at the point (1/32v0, 1/64 (v0)^2)

1/32v0 represents the time when the firework is at its maximum height and 1/64 (v0)^2 is this maximum height

Answer 2

Answer with explanation:

The equation representing the height of the firework is

`h=-16t^2+v_(0)t+h_(0)\\\\h=-16(t^2-(v_(0)t)/(16)-(h_(0))/(16))\\\\h=-16[(t-(v_(0))/(32))^2-((v_(0))/(32))^2-(h_(0))/(16)]\\\\h-16[((v_(0))/(32))^2+(h_(0))/(16)]=-16[(t-(v_(0))/(32))]^2\\\\ \text{This is a function in t and h}\\\\ \text{Vertex}=((v_(0))/(32),16[((v_(0))/(32))^2+(h_(0))/(16))])`

To determine the extreme of the function , we will differentiate it once

`h=-16t^2+v_(0)t+h_(0)\\\\(dh)/(dt)=-32 t+v_(0)\\\\\text{Put},(dh)/(dt)=0\\\\\rightarrow -32 t+v_(0)=0\\\\\rightarrow t=(v_(0))/(32)\\\\\text{Extreme value}=-16*[(v_(0))/(32)]^2+v_(0)*(v_(0))/(32)+h_(0)\\\\f(t)=-16*[(v_(0))/(32)]^2+((v_(0))^2)/(32)+h_(0)\\\\(d^2h)/(dt^2)=-32\\\\\text{Showing that function attains maximum at this point}`

The Extreme Value represents maximum height obtained by Firework.