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Calculus anti derivative problem?

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The projected rate of increase in enrollment at a new college is estimated by: 

dE/dt=7,000(t+16)^(-3/2) 

where E(t) is the projected enrollment in t years. If the enrollment is 1,000 now (t=0), find the projected enrollment 9 years from now.

asked Dec 13, 2014 in CALCULUS by anonymous

1 Answer

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The projected rate of increase in enrollment at a new college is : dE / dt = 7000(t + 16)(- 3/2).

⇒ dE = 7000(t + 16)(- 3/2) * dt

Let, u = t + 16

du = dt.

⇒ dE = 7000(u)(- 3/2) * du

ʃ dE = ʃ [7000(u)(- 3/2) * du]

E = 7000ʃ [(u)(- 3/2) du]

Apply formula : ʃ xn dx = xn + 1 /(n + 1) + C.

E = 7000[u(- 3/2 + 1)] / (- 3/2 + 1)] + C

E = 7000[u(- 1/2)] / (- 1/2)] + C

E = - 2 * 7000 * u(- 1/2) + C

E = - 14000 * u(- 1/2) + C

Put, u = t + 16.

E = - 14000(t + 16)(- 1/2) + C.

At, t = 0, E = 1000.

- 14000(0 + 16)(- 1/2) + C = 1000

C = 1000 + 14000(16)(- 1/2)

C = 1000 + 14000(4)(- 1)

C = 1000 + (14000/4)

C = 1000 + 3500

C = 4500.

Therefore, E = - 14000(t + 16)(- 1/2) + 4500.

At, t =  9,

E = - 14000(9 + 16)(- 1/2) + 4500

E = - 14000(25)(- 1/2) + 4500

E = - 14000(5)(- 1) + 4500

E = (- 14000/5) + 4500

E = - 2800 + 4500

E = 1700.

Therefore, the projected enrollment 9 years from now is 1700.

answered Dec 13, 2014 by lilly Expert

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