=

(c) Show that the maximum value of xn (t) for n > 0 is

Mn = xn (2n) = n" e "/n!. (d) Conclude from Stirling's

approximation n! ≈n"e" √2лn that M₁,~ (2лn)-¹/².

41. A 30-year-old woman accepts an engineering position

with a starting salary of $30,000 per year. Her salary

S(t) increases exponentially, with S(t) 30e¹/20 thou-

sand dollars after t years. Meanwhile, 12% of her salary

is deposited continuously in a retirement account, which

accumulates interest at a continuous annual rate of 6%.

(a) Estimate AA in terms of At to derive the differential

equation satisfied by the amount A(t) in her retirement

account after t years. (b) Compute A (40), the amount

available for her retirement at age 70.

42. Suppose that a falling hailstone with density &

from rest with negligible radius r = 0. Thereafter its ra-

dius is r = kt (k is a constant) as it grows by accretion

during its fall. Use Newton's second law-according to

= 1 starts