In a chemical reaction, substance A combines with substance B to form substance Y. At the start of the reaction, the quantity of A present is a grams, and the quantity of B present is b grams. Assume a> and <img src=. At time t seconds after the start of the reaction, the quantity of Y present is y grams. For certain types of reactions, the rate of the reaction, in grams/sec, is given by
\hbox{Rate}=k(a-y)(b-y),
where k is a positive constant. 1. Sketch a graph of the rate against y. For what values of y is the rate nonnegative?
a) [0,a]
b) 0
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If we expect the rate to be nonnegative, we must have 0 ≤ y ≤ a and 0 ≤ y ≤ b.
Since we assume a < b, we restrict y to 0 ≤ y ≤ a.
In fact, the expression for the rate is nonnegative for y greater than b, but these
values of y are not meaningful for the reaction.
(b) From the graph, we see that the maximum rate occurs when y = 0; that is, at the
start of the reaction.
a) y E ________(0,a)________
b) y = ___0____
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To sketch a graph of the rate against y, we can analyze the factors in the equation: rate = k(a-y)(b-y).
First, let's consider the factors (a-y) and (b-y). From the given information, we know that a > y and b > y.
When y is less than both a and b, both factors (a-y) and (b-y) are positive. As y increases, the factors decrease until y reaches either a or b. At y = a, the factor (a-y) becomes zero, and at y = b, the factor (b-y) becomes zero.
Now, consider the factor (a-y)(b-y). When both factors are positive (y < a and y < b), the product is positive. When one factor becomes zero (y = a or y = b) while the other factor is still positive, the product remains zero. When both factors become zero at y = a and y = b, the product remains zero.
Based on this analysis, we can conclude the following:
1. When y < a and y < b, the rate is nonnegative (positive or zero).
2. When y = a or y = b, the rate is zero.
3. When y > a or y > b, the rate is negative.
To summarize, the rate is nonnegative for values of y that are less than both a and b, i.e., y < a and y < b.