Q. 19 - Precalculus Enhanced with Graphing Utilities

Question

Chemical Reactions: A self-catalytic chemical reaction results in the formation of a compound that causes the formation ratio to increase. If the reaction rate $V$ is modeled by $V\left(x\right)=kx(a-x), 0\le x\le a$

where $k$ is a positive constant, $a$ is the initial amount of the compound and $x$ is the variable of the compound for what value of $x$ is the reaction rate is maximum?

Step-by-Step Solution

Answer

The value of $x$ is $\frac{a}{2}$, the reaction rate is maximum.

Step 1. Given Information

It is given that the reaction rate $V$ is modeled by $V\left(x\right)=kx(a-x), 0\le x\le a$, we need to determine the value of $x$ in which the reaction rate is maximum.

Step 2. Simplify

$V\left(x\right)=kx(a-x)$

       $=-k{x}^2+kax$

The vertex of the function is calculated by the formula,

$x=\frac{-ka}{-2·(-k)}$

$x=\frac{a}{2}$

For value of $x=\frac{a}{2}$ the reaction rate is maximum.