Question

use the given quadratic function to answer questions about the situation it models.

1. a cannonball is fired into the air with an initial vertical velocity of 128 feet per second. the release point is 6 feet above the ground. the function ( h = - 16t^{2}+128t + 6 ) represents the height ( h ) (in feet) of the cannonball after ( t ) seconds.

(a) find the height of the cannonball each second after it is fired.
(b) use the graph of the model to determine how long the cannonball is in the air.

Explanation:

Step1: Calculate height at each second for part (a)

For \(t = 0\):
\(h=-16(0)^2 + 128(0)+6=6\)
For \(t = 1\):
\(h=-16(1)^2 + 128(1)+6=-16 + 128+6 = 118\)
For \(t = 2\):
\(h=-16(2)^2 + 128(2)+6=-64+256 + 6=198\)
For \(t = 3\):
\(h=-16(3)^2 + 128(3)+6=-144+384+6 = 246\)
For \(t = 4\):
\(h=-16(4)^2 + 128(4)+6=-256+512+6=262\)
For \(t = 5\):
\(h=-16(5)^2 + 128(5)+6=-400+640+6=246\)
For \(t = 6\):
\(h=-16(6)^2 + 128(6)+6=-576+768+6=198\)
For \(t = 7\):
\(h=-16(7)^2 + 128(7)+6=-784+896+6=118\)
For \(t = 8\):
\(h=-16(8)^2 + 128(8)+6=-1024+1024+6=6\)

Step2: Find time when \(h = 0\) for part (b)

Set \(h=-16t^2 + 128t + 6=0\)
Use the quadratic formula \(t=\frac{-b\pm\sqrt{b^2 - 4ac}}{2a}\) where \(a=-16\), \(b = 128\), \(c = 6\)
\(t=\frac{-128\pm\sqrt{128^2-4(-16)(6)}}{2(-16)}=\frac{-128\pm\sqrt{16384 + 384}}{-32}=\frac{-128\pm\sqrt{16768}}{-32}=\frac{-128\pm129.5}{-32}\)
We take the positive root \(t=\frac{-128 + 129.5}{-32}\approx8.0\) (we can also observe from the symmetry of the parabola \(y=-16t^2+128t + 6\), the axis of symmetry \(t=-\frac{b}{2a}=-\frac{128}{2(-16)} = 4\), and since the function is symmetric about \(t = 4\) and \(h(0)=h(8)=6\), when \(h = 0\) (ground level), using the quadratic formula or graph - observing (assuming the graph is a parabola opening downwards \(y=-16t^2+128t + 6\)), the non - zero root (since \(t>0\) when it hits the ground after being fired) is approximately \(t = 8\) seconds

Answer:

(a)

\(t\) (seconds)	\(h\) (feet)
\(1\)	\(118\)
\(2\)	\(198\)
\(3\)	\(246\)
\(4\)	\(262\)
\(5\)	\(246\)
\(6\)	\(198\)
\(7\)	\(118\)
\(8\)	\(6\)

(b) The cannonball is in the air for approximately \(8\) seconds.