Question

1. a cannonball is fired from a cliff. the height (h) of the cannonball (in feet) as a function of time in seconds. can be modeled by h(t)=-16t² + 64t + 260. how long until the cannonball hits the ground? show set up, then solve.

Explanation:

Step1: Set height to 0

$-16t^{2}+64t + 260=0$

Step2: Factor out GCF (-4)

$-4(4t^{2}-16t - 65)=0$

Step3: Divide both sides by -4

$4t^{2}-16t - 65 = 0$

Step4: Use quadratic formula $t=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$

Here $a = 4$, $b=-16$, $c=-65$. First calculate the discriminant $\Delta=b^{2}-4ac=(-16)^{2}-4\times4\times(-65)=256 + 1040=1296$

Step5: Find t values

$t=\frac{16\pm\sqrt{1296}}{8}=\frac{16\pm36}{8}$
We get two solutions: $t_1=\frac{16 + 36}{8}=\frac{52}{8}=6.5$ and $t_2=\frac{16-36}{8}=\frac{-20}{8}=-2.5$

Step6: Reject negative time

Since time cannot be negative in this context, we discard $t = - 2.5$

Answer:

$t = 6.5$ seconds