Question 263217
Laboratories it was found that the number of components
assembled per hour by the average worker t hours after
starting work could be modeled by the formula
N(t)=-3t^3 + 23t^2 +8t.
:
a) Rewrite the formula by factoring the right-hand side
completely.
n(t) = -3t^3 + 23t^2 + 8t 
factor out -t, changing signs makes it easier to factor
n(t) = -t(3t^2 - 23t - 8) 
n(t) = -t(3t + 1)(t - 8)
:
b) Use the factored version of the formula to find N(3).
n(t) = -t(3t + 1)(t - 8)
n(3) = -3(3(3) + 1)(3 - 8)
n(3) = -3(10)(-5)
n(3) = +150 units
:
c) Use the accompanying graph to estimate the time at
which the workers are most efficient.
{{{ graph( 300, 200, -4, 10, -50, 320, -3x^3+23x^2+8x) }}}
Looks to be about 5.25 hrs, wouldn't you think?
:
d) Use the accompanying graph to estimate the
maximum number of components assembled per
hour during an 8-hour shift. 
:
at 5.25 hrs it looks like 250 units per hr, however we can calculate that
n(t) = -t(3t + 1)(t - 8)
n(5.25) = -5.25(3(5.25) + 1)(5.25-8)
n(5.25) = -5.25(16.75)(-2.75)
n(5.25) = 241.8 ~ 240 units/hr would be closer