Fireworks are fired from the roof of a 100-foot building and travel 84 feet per second. The equation h= -16^2+84t+100 models the height h of the fireworks at any given time t seconds.

Question

A: How long are the fireworks in the air?
B: How high did the fireworks get?
C: How high were the fireworks 2 seconds after they left the roof?

Answer 1

A: To find how long the fireworks are in the air, we need to find the time it takes for the height to reach 0 (when the fireworks hit the ground).
Setting h = 0 in the equation gives us:
0 = -16t^2 + 84t + 100
Rearranging this equation, we get:
16t^2 - 84t - 100 = 0
Dividing by 4 to simplify, we get:
4t^2 - 21t - 25 = 0
Factoring:
(4t + 5)(t - 5) = 0
t = -5/4 or t = 5
Since time can't be negative in this context, the fireworks are in the air for 5 seconds.

B: To find how high the fireworks get, we need to find the maximum height which occurs at the vertex of the parabolic equation h = -16t^2 + 84t + 100.
The formula for the x-coordinate of the vertex of a parabola is given by x = -b/2a. In our case, a = -16, b = 84.
So, x = -84 / (2*(-16)) = 84/32 = 2.625 seconds.
To find the maximum height:
h = -16(2.625)^2 + 84(2.625) + 100 = 220.5 feet
The fireworks reach a maximum height of 220.5 feet.

C: To find how high the fireworks were 2 seconds after they left the roof, we substitute t = 2 into the equation:
h = -16(2)^2 + 84(2) + 100
h = -64 + 168 + 100 = 204 feet
The fireworks were 204 feet high 2 seconds after they left the roof.