a.) A study prepared for a Sunbelt town\'s chamber of commerce projected that th

Question

a.) A study prepared for a Sunbelt town's chamber of commerce projected that the town's population in the next 3 years will grow according to the rule
P(t) = 59,000 + 28t3/2 + 25t
where P(t) denotes the population t months from now. How fast will the population be increasing 9 months and 16 months from now?
9 months     people/month
16 months     people/month

b.) A straight line perpendicular to and passing through the point of tangency of the tangent line is called the normal to the curve at that point. Find an equation of the tangent line and the normal to the curve
y = x3 3x + 2 at the point (3, 20).

tangent line     y =
normal line     y =

c.) The altitude (in feet) of a rocket t sec into flight is given by
s = f(t) = 2t3 + 112t2 + 640t + 1 (t 0)
(1) Find an expression v for the rocket's velocity at any time t.

v =  
(2) Compute the rocket's velocity when t = 0, 20, 40, and 60.
t = 0 ft/sec
t = 20 ft/sec
t = 40 ft/sec
t = 60 ft/sec

Explanation / Answer

a) given

P(t) = 59,000 + 28t3/2 + 25t

differentiate with respect to time t ,d/dx xn=nxn-1

rate of change of population dP/dt = 0 + 28(3/2)t(3/2)-1 + 25*1

dP/dt = 42t(3/2)-1 + 25

dP/dt = 42t(1/2) + 25

9 months from now

dP/dt = 42*9(1/2) + 25

dP/dt = 42*3 + 25

dP/dt = 151 people per month

population will be increasing 9 months from at  151 people per month

16 months from now

dP/dt = 42*16(1/2) + 25

dP/dt = 42*4 + 25

dP/dt = 193 people per month

population will be increasing 16 months from at  193 people per month

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