for his schools science fair, brad made a bot...
for his schools science fair, brad made a bottle rocket powered by the pressure of the carbonated soda in the bottle. he launches the rocket, and it runs out of soda at a height of 9 meters. at that moment, the rocket is moving upward with a velocity of 7 meters per second. it continues to travel upward before falling to the ground. to the nearest tenth of a second, how long does it take the rocket to hit the ground after running out of soda? hint: use the formula h = -4.9t² + vt + s.
Answer
# Explanation:
## Step1: Identify the values for the formula
We know that when the rocket hits the ground, $h = 0$, the initial - velocity $v=7$ m/s, and the initial height $s = 9$ m. The formula is $h=-4.9t^{2}+vt + s$. Substituting the values, we get the quadratic equation $0=-4.9t^{2}+7t + 9$.
## Step2: Use the quadratic formula
The quadratic formula for a quadratic equation $ax^{2}+bx + c = 0$ is $t=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$. For the equation $-4.9t^{2}+7t + 9 = 0$, we have $a=-4.9$, $b = 7$, and $c = 9$.
First, calculate the discriminant $\Delta=b^{2}-4ac=(7)^{2}-4\times(-4.9)\times9=49 + 176.4=225.4$.
Then, $t=\frac{-7\pm\sqrt{225.4}}{2\times(-4.9)}=\frac{-7\pm15.013}{-9.8}$.
## Step3: Find the positive root
We have two solutions for $t$:
$t_1=\frac{-7 + 15.013}{-9.8}=\frac{8.013}{-9.8}\approx - 0.82$ (rejected since time cannot be negative).
$t_2=\frac{-7-15.013}{-9.8}=\frac{-22.013}{-9.8}\approx2.2$.
# Answer:
$2.2$