Business Calculus

Topic: Indefinite Integrals

The rate of disbursement of dQ/dt of a $9 million federal grant is proportional to the square of 100 - t, where t is the time (in days, 0 ≤ t ≤ 100) and Q is the amount that remains to be disbursed.

Find the amount that remains to be disbursed after 40 days.
*Assume the grant will be disbursed after 100 days.*

$1.944 million

Step-by-step explanation:

dQ/dt = k(100-t)²

Q = -k(100-t)³/3 + c

t = 100, Q = 0

t = 0, Q = 9000000

0 = -k(100-100)³/3 + c

So c = 0

Q = -k(100-t)³/3

9000000 × 3 = -k(100)³

27000000 = -1000000k

k = -27

Q = 27(100-t)³/3

Q = 9(100-t)³

t = 40

Q = 9(60)³

Q = $1944000

Q = $1.944 million

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