Please apply the corrections to this problem.

$6.$ Suppose two substances in a chemical reaction produce a new substance, known as the reaction

product, and the properties of the chemical reaction lead to the following equation

Z $1$

$(80$ x$)(50$ x$)$ dx $=$

Z

k dt$,$

where k is a constant, x$($t$)$ is the number of kg of the reaction substance t minutes after the

reaction started, and the initial amounts of the two reacting substances are $80$ kg and $50$ kg$,$

respectively.

$($a$)$ Integrate on each side, using an appropriate method of integration, to get an equation

relating a function of x to a function of t$.$

$($b$)$ Solve your equation from part $($a$)$ to find a formula for x$($t$),$ using the initial condition

x$(0)=0.($Your formula should involve exponential functions.$)$

$($c$)$ How much reaction product x$($t$)$ is present after $15$ minutes, assuming that x$(10)=20?$

Explain