When metallic tin is kept below 13.2°C, it slowlybecomes brittle and crumbles to a gray powder. Tin objects eventuallycrumble to this gray powder spontaneously if kept in acold climate for years. The Europeans who saw tin organ pipes intheir churches crumble away years ago called the change tin pestbecause it seemed to be contagious, and indeed it was, for the graypowder is a catalyst for its own formation.A catalyst for a chemical reaction is a substance that controls therate of reaction without undergoing any permanent change in itself.An autocatalytic reaction is one whose product is a catalyst for itsown formation. Such a reaction may proceed slowly at first if theamount of catalyst present is small and slowly again at the end,when most of the original substance is used up. But in between,when both the substance and its catalyst product are abundant, thereaction proceeds at a faster pace.In some cases, it is reasonable to assume that the rate y = dx/dtof the reaction is proportional both to the amount of the originalsubstance present and to the amount of product. That is, y may beconsidered to be a function of x alone, andy = kx(a - x) = kax - kx2,wherex = the amount of producta = the amount of substance at the beginningk = a positive constant.At what value of x does the rate y have a maximum? What is themaximum value of y?

When metallic tin is kept below 13.2°C, it slowly
becomes brittle and crumbles to a gray powder. Tin objects eventually
crumble to this gray powder spontaneously if kept in a
cold climate for years. The Europeans who saw tin organ pipes in
their churches crumble away years ago called the change tin pest
because it seemed to be contagious, and indeed it was, for the gray
powder is a catalyst for its own formation.
A catalyst for a chemical reaction is a substance that controls the
rate of reaction without undergoing any permanent change in itself.
An autocatalytic reaction is one whose product is a catalyst for its
own formation. Such a reaction may proceed slowly at first if the
amount of catalyst present is small and slowly again at the end,
when most of the original substance is used up. But in between,
when both the substance and its catalyst product are abundant, the
reaction proceeds at a faster pace.
In some cases, it is reasonable to assume that the rate y = dx/dt
of the reaction is proportional both to the amount of the original
substance present and to the amount of product. That is, y may be
considered to be a function of x alone, and
y = kx(a - x) = kax - kx²,
where
x = the amount of product
a = the amount of substance at the beginning
k = a positive constant.
At what value of x does the rate y have a maximum? What is the
maximum value of y?