3. The amount of a chemical increases at a rate equal to the product of elapsed time (in minutes) and the amount of the chemical. If the initial amount of the chemical is 10 units, what is the number of units at 4 minutes?
A. 14 |
B. 10 + e^{8} |
C. 10 + e^{16} |
D. 10e^{8} |
E. 10e^{16} |
Solution: The answer is D
Let A(t) = the amount of a chemical at any time t (in minutes).
Since the amount of a chemical increases at a rate equal to the product of elapsed time and the amount of the chemical, we have the following differential equation,
Multiplying both sides by dt, dividing by A(t) and integrating both sides yields the following,
whose solution is given by,
We can solve for C by using the given fact that A(0)=10. So,
Hence,
Lastly, solving for the number of units of the chemical at 4 minutes. Simply let t = 4 and substitute this result in the above equation, so that,