Can someone PLEASE help me with this problem??
In a study of worker efficiency at Wong
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)��3t3 � 23t2 � 8t.
a) Rewrite the formula by factoring the right-hand side
completely.
b) Use the factored version of the formula to find N(3).
c) Use the accompanying graph to estimate the time at
which the workers are most efficient.
d) Use the accompanying graph to estimate the
maximum number of components assembled per
hour during an 8-hour shift.
a) To factor the right-hand side of the formula, we can apply the process of factoring by grouping. The given formula is:
N(t) = 3t^3 - 23t^2 + 8t
Step 1: Factor out the greatest common factor (GCF)
N(t) = t(3t^2 - 23t + 8)
Step 2: Factor the quadratic expression inside the parentheses
N(t) = t(3t - 1)(t - 8)
So, the factored version of the formula is N(t) = t(3t - 1)(t - 8).
b) To find N(3), we substitute t = 3 into the factored formula we obtained in part a:
N(3) = 3(3(3) - 1)(3 - 8)
N(3) = 3(9 - 1)(-5)
N(3) = 3(8)(-5)
N(3) = -120
Therefore, N(3) = -120.
c) The accompanying graph can be used to estimate the time at which the workers are most efficient. Look for the highest point on the graph, which represents the maximum value of N(t) (the number of components assembled per hour). The x-coordinate (time) at this highest point represents the estimated time at which the workers are most efficient.
d) Similarly, the accompanying graph can be used to estimate the maximum number of components assembled per hour during an 8-hour shift. Find the highest point on the graph within the range of 8 hours, and the y-coordinate (number of components assembled per hour) at this point represents the estimated maximum number of components assembled during the 8-hour shift.