is proportional to the amount present. In a certain reaction, the change of the substance satisfies the differential equation𝑑𝑦𝑑𝑡 = -0.64ywhere y is measured in grams and t is measured in hours. If there are 90 grams of the substance when t = 0, how many grams will be left after the first hour?
Question
is proportional to the amount present. In a certain reaction, the change of the substance satisfies the differential equation𝑑𝑦𝑑𝑡 = -0.64ywhere y is measured in grams and t is measured in hours. If there are 90 grams of the substance when t = 0, how many grams will be left after the first hour?
Solution
The given differential equation is a first order linear differential equation, which represents a model of exponential decay. The general solution to such an equation is given by:
y(t) = y(0) * e^(kt)
where:
- y(t) is the amount of substance at time t,
- y(0) is the initial amount of the substance,
- k is the rate constant,
- e is the base of the natural logarithm (approximately equal to 2.71828).
In this case, we have:
- y(0) = 90 grams (the initial amount),
- k = -0.64 per hour (the rate constant).
We want to find y(1), the amount of substance after 1 hour. Substituting the given values into the equation, we get:
y(1) = 90 * e^(-0.64*1) = 90 * e^(-0.64) = 90 * 0.527 = 47.43 grams
So, after the first hour, there will be approximately 47.43 grams of the substance left.
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