$\text { The rate of disbursement } d Q / d t \text { of a } 2 \text { million dollar federal grant is proportional to }$ the square of $100 - t$. Time $t$ is measured in days $( 0 \leq t \leq 100 )$ and $Q$ is the amount that remains to be disbursed. Find the amount that remains to be disbursed after 30 days. Assume that all the money will be disbursed in 100 days.
A) $\$ 686,000$
B) $\$ 50,700$
C) $\$ 101,400$
D) $\$ 33,800$
E) $\$ 1,029,000$

Question

Quizplus · Accepted Answer

The rate of disbursement is given by $ \frac{dQ}{dt} = -k(100-t)^2 $. Integrating this with the initial condition $ Q(0) = 2,000,000 $ and $ Q(100) = 0 $, we find $ k $. Solving for $ Q(30) $, we get approximately \$686,000.

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