Question

Jayden's best friend brought him 10 packs of bottle rocket fireworks for his

birthday that he launched for July 4th weekend. The height of the fireworks
can be modeled by the function h(x) = -x² + 10.5x + 2 where h is the height
in feet and t is the time in seconds.

b. Determine the maximum height of the fireworks and the time it takes
to reach that height (Round to the nearest tenth). Use
mathematics to justify your answer. Use proper units. (2 points)

c. Determine the length of time that the bottle rocket was in the air. Use
mathematics to justify your answer (Round to the nearest tenth).
Use proper units. (2 points)

Answer 1

t = time in seconds after the ball was thrown

h(t) = the height of the ball after t seconds.

h(t)=−16t2+64t+80

Question (1): What is the maximum height of the ball?

Remember that the graph of the function is a parabola open down because the leading coefficient is negative. Therefore, to get the maximum height we just have to find the vertex of this parabola. Since the function is in standard form h(t) = at2+bt+c, the formula for getting the vertex is:

V(h,k) = (-b/(2a), h(-b/(2a)))

-b/(2a) = -64/(2*(-16)) = 2 seconds (the time it reaches the highest point.)

h(-b/(2a)) = h(2) = -16(2)2 + 64(2) + 80 = 144 feet (Maximum height)

Question (2): How many seconds does it take until the ball hits the ground?

If it hits the ground, it means the height is zero. h(t)=0.

0= -16t2 + 64t + 80

Factor:

0 = -16(t2 - 4t - 5)

0 = -16(t - 5) (t + 1)

t-5 =0 or t+1=0

We eliminate t+1=0 because of the negative value of t. Time should always be positive in this case.

t = 5 seconds to hit the ground.