Question

a rocket is launched into the air. the projectile motion of the rocket can be modeled using ( h(t)=112t - 16t^{2} ), where ( t ) is the time since launch in seconds and ( h(t) ) is the height of the rocket at time ( t ). when will the rocket be 196 feet in the air? after 2.0 seconds after 3.5 seconds after 4.5 seconds after 7.0 seconds

Explanation:

Step1: Set up the equation

We want to find \(t\) when \(h(t)=196\). So we set \(112t - 16t^{2}=196\).
Rearrange it to \(16t^{2}-112t + 196 = 0\). Divide through by \(4\) to simplify: \(4t^{2}-28t + 49 = 0\).

Step2: Use the quadratic formula

For a quadratic equation \(ax^{2}+bx + c = 0\) (\(a = 4\), \(b=-28\), \(c = 49\)), the quadratic formula is \(t=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}\).
First, calculate the discriminant \(\Delta=b^{2}-4ac=(-28)^{2}-4\times4\times49=784 - 784=0\).
Then \(t=\frac{-(-28)\pm\sqrt{0}}{2\times4}=\frac{28}{8}=3.5\).

Answer:

after \(3.5\) seconds