Question

your chemistry teacher conducts a lab where you launch a coca - cola bottle using an acid - base reaction between vinegar and baking soda. the following equation represents the height of the bottle after being launched.
h(t) = - 16t² + 32t + 2
a. find the maximum height that the rocket reaches

Explanation:

Step1: Identify the quadratic function form

The height function is \(h(t)=-16t^{2}+32t + 2\), which is in the form \(y = ax^{2}+bx + c\) (\(a=-16\), \(b = 32\), \(c = 2\)).

Step2: Find the time \(t\) at which the maximum occurs

For a quadratic function \(y=ax^{2}+bx + c\), the \(x\) - coordinate (in this case \(t\)) of the vertex (where maximum occurs since \(a=-16<0\)) is given by \(t=-\frac{b}{2a}\).
Substitute \(a=-16\) and \(b = 32\) into the formula:
\(t=-\frac{32}{2\times(-16)}=\frac{-32}{-32}=1\).

Step3: Find the maximum height

Substitute \(t = 1\) into the height function \(h(t)\):
\(h(1)=-16\times(1)^{2}+32\times(1)+2\).
First, calculate \(-16\times(1)^{2}=-16\), \(32\times(1)=32\).
Then \(h(1)=-16 + 32+2\).
\(h(1)=18\).

Answer:

The maximum height that the rocket reaches is \(18\) units.