Question

given quadratic function to answer questions about the situation it models.
the height h (in feet) of a projectile with an initial vertical velocity of 96 ft/sec is modeled by the function
$h=-16t^{2}+96t$, where t is the time, in seconds.
(a) find the height of the projectile each second after it is fired into the air.
step 1: make a table of values starting with t = 0. continue the table until a function value is
negative.
step 2: interpret the values in the table.
(b) use the graph of the model to determine how long the
projectile is in the air.

Explanation:

Step1: Calculate height for \(t = 0\)

Substitute \(t = 0\) into \(h=-16t^{2}+96t\).
\(h=-16\times(0)^{2}+96\times(0)=0\)

Step2: Calculate height for \(t = 1\)

Substitute \(t = 1\) into \(h=-16t^{2}+96t\).
\(h=-16\times(1)^{2}+96\times(1)=-16 + 96=80\)

Step3: Calculate height for \(t = 2\)

Substitute \(t = 2\) into \(h=-16t^{2}+96t\).
\(h=-16\times(2)^{2}+96\times(2)=-64+192 = 128\)

Step4: Calculate height for \(t = 3\)

Substitute \(t = 3\) into \(h=-16t^{2}+96t\).
\(h=-16\times(3)^{2}+96\times(3)=-144 + 288=144\)

Step5: Calculate height for \(t = 4\)

Substitute \(t = 4\) into \(h=-16t^{2}+96t\).
\(h=-16\times(4)^{2}+96\times(4)=-256+384 = 128\)

Step6: Calculate height for \(t = 5\)

Substitute \(t = 5\) into \(h=-16t^{2}+96t\).
\(h=-16\times(5)^{2}+96\times(5)=-400+480 = 80\)

Step7: Calculate height for \(t = 6\)

Substitute \(t = 6\) into \(h=-16t^{2}+96t\).
\(h=-16\times(6)^{2}+96\times(6)=-576+576=0\)

Step8: Calculate height for \(t = 7\)

Substitute \(t = 7\) into \(h=-16t^{2}+96t\).
\(h=-16\times(7)^{2}+96\times(7)=-784 + 672=-112\)

The table of values:

\(t\), seconds	\(0\)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)	\(6\)	\(7\)

For part (b), the projectile is in the air when \(h>0\). From the table and the quadratic function \(h=-16t^{2}+96t=-16t(t - 6)\), the roots of the quadratic equation \(h = 0\) are \(t = 0\) (launch time) and \(t = 6\) (landing time).

Answer:

(a) The table of values:

\(t\), seconds	\(0\)	\(1\)	\(2\)	\(3\)	\(4\)	\(5\)	\(6\)	\(7\)

(b) The projectile is in the air for \(6\) seconds.