Question

if a cannon - ball is fired from a height of 8.8 meters above the ground, at a certain angle, the height of the cannon - ball above the ground, h, in meters, at time, t, in seconds, is found by the function h(t)=-4.9t² + 24.5t + 8.8. find the time it takes for the cannon - ball to strike the ground. (type an integer or a decimal. round to the nearest hundredth as needed.)

Explanation:

Step1: Set height function to 0

We want to find when the cannon - ball hits the ground, so we set $h(t)=0$. The function is $h(t)=-4.9t^{2}+24.5t + 8.8$. So, $-4.9t^{2}+24.5t + 8.8 = 0$.

Step2: Use quadratic formula

For a quadratic equation $ax^{2}+bx + c = 0$, the solutions are given by $t=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. Here, $a=-4.9$, $b = 24.5$, and $c = 8.8$. First, calculate the discriminant $\Delta=b^{2}-4ac=(24.5)^{2}-4\times(-4.9)\times8.8=600.25+172.48 = 772.73$.

Step3: Calculate t values

$t=\frac{-24.5\pm\sqrt{772.73}}{2\times(-4.9)}=\frac{-24.5\pm27.8}{-9.8}$. We have two solutions for $t$: $t_1=\frac{-24.5 + 27.8}{-9.8}=\frac{3.3}{-9.8}\approx - 0.34$ and $t_2=\frac{-24.5-27.8}{-9.8}=\frac{-52.3}{-9.8}\approx5.34$. Since time cannot be negative in this context, we discard the negative value.

Answer:

$5.34$