U.S. Nursing Shortage

Question

The demand for a certain profession between 2000 and 2015 is estimated to be

D(t) = 0.0004t² + 0.0235t + 5      (0 ≤ t ≤ 15)

where D(t) is measured in millions and t = 0

 corresponds to the year 2000. The supply of these professionals over the same time period is estimated to be

S(t) = −0.0013t² + 0.032t + 4.8      (0 ≤ t ≤ 15)

where S(t) is also measured in millions.

Find an expression G(t) giving the gap between the demand and supply of these professionals over the period in question.

G(t) =

and

Find the relative extrema of G. (Round your answers to four decimal places. If an answer does not exist, enter DNE.)

Relative minimum (t, G(t)) =

Relative maximum (t, G(t))=

Answer 1

S(t) =-.0013t^2 + .032t +4.8

D(t) = .0004t^2 + .0235t + 5

Nursing Shortage = G(t) = D(t)-S(t) =(.0004+ .0013)t^2 +(.0234 -.032)t + 5-4.8

=.0017t^2 -.00862t - .2

initially, in the year 2000, at time t=0, the shortage = 0.2 million= 200,000

in 2015, at time t=15, the shortage = .017(225) - .0172(15) +.2 = 3.825 -.258 +.2

= 3.767 million = 3,767,000

take the derivative and set = 0

G'(t) = .0034t -.00862 = 0

t= .00862/.0034 = about 2.69375= the year 2002

with a Nursing shortage = .0017(2.69375)^2 - .00862(2.69375) -.2 = .01233569 - .023220125 +.2= .189115565

= 189,116

Graphically it's an upward opening parabola with vertex (2.6938, 0.1891) in the year 2002

the global maximum in the interval (0,15) or (2000, 2015) at 2015. 2002 has the smallest nursing shortage

the global and local or relative minimum is in the year 2002 with 0.1891 million rounded to 4 decimal places

(t, G(t) = (2.6938, 0.1891)

there is no relative maximum in the 15 year interval. DNE. It does not exist

Answer 2

G(t) = S(t) - D(t)

Determine the difference

Solve G'(t) = 0