SOLUTION: I would appreciate assistance with how this problem is completed. In a study of worker efficiency at Wong Laboratories it was found that the number of components assembled pe

Question 292578: I would appreciate assistance with how this problem is completed.
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)=-3t^3+23t^2+8t.
a) Rewrite the formula by factoring the right-hand side
completely.
b) Use the factored version of the formula to find N(3).
Found 2 solutions by nerdybill, Theo:
Answer by nerdybill(7384) (Show Source): You can put this solution on YOUR website!
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
N(t)=t(-3t^2+23t+8)
N(t)=t(-3t^2+24t-t+8)
N(t)=t((-3t^2+24t)-(t-8))
N(t)=t(3t(-t+8)-(t-8))
N(t)=t(-3t(t-8)-(t-8))
N(t)=t(t-8)(-3t-1)
.
b) Use the factored version of the formula to find N(3)
N(t)=t(t-8)(-3t-1)
N(3)=3(3-8)(-3(3)-1)
N(3)=3(-5)(-9-1)
N(3)=3(-5)(-10)
N(3)=(-15)(-10)
N(3)=150

Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
Equation is:

-3t^3+23t^2+8t

Factor by "t" to get:

t * (-3t^2 + 23t + 8)

Set this equation equal to 0 to find the roots.

t * (-3t^2 + 23t + 8) = 0

Factor (-3t^2 + 23t + 8) to get:

t * (3t+1) * (-t+8) = 0

This makes t = 0 or (3t+1) = 0 or (-t+8) = 0

If 3t+1 = 0, then t = -1/3

If -t+8 = 0, then t = 8

or t = 0.

Your 3 possible roots are:

t = -1/3
t = 8
t = 0

Plug these into your original equation to see if they are good.

Your original equation is:

-3t^3+23t^2+8t

When t = 0, this equation becomes 0.

When t = -1/3, this equation becomes -3*(-1/3)^3 + 23*(-1/3)^2 + 8*(-1/3).

This becomes .11111111 + 2.55555555 - 2.66666666 which becomes 0.

When t = 8, this equation becomes -3*8^3 + 23*8^2 + 8*8.

This becomes -1536 + 1472 + 64 which becomes 0.

All 3 roots are good.

Our factored equation is:

N(t) = t * (3t+1) * (-t+8) = 0

When we substitute 3 for t, we get:

N(3) = 3 * (3*3+1) * (-3+8) which becomes:

N(3) = 3 * 10 * 5 which becomes:

N(3) = 150

Plug 3 into the original equation of -3t^3+23t^2+8t and you get -3*(3^3) + 23*(3^2) + 8*3 which becomes -3*27 + 23*9 + 24 which becomes -81 + 207 + 24 which becomes 150.

The factoring is good and the answer is N(3) = 150

I determined the factoring as follows:

I knew that 1*8 = 8 or 2*4 = 8

I tried 1*8 first.

I knew that 3*8 = 24 and -1*8 = -1 so I figured that the factors had to be something like (3t+1) * (-t+8)

When you multiply these together, you get:

(3t+1) * (-t+8) =
(3t * (-t+8)) + (1 * (-t+8)) =
(-3t^2 + 24t) + (-t + 8) =
-3t^2 + 24t -t + 8 =
-3t^2 + 23t + 8 which makes them the correct factors.

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