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The number 'N' of cars produced at a certain factory in 1 day after 't' hours of operation is given by N(t)=100t-5t^2, 0< or equal t < or equal 10. If the cost 'C' (in dollars) of producing 'N' cars is C(N)=15,000+8000N, find the cost 'C' as a function of the time 't' of operation of the factory

Then Interpret C(t) when t=5 hours as a new function.



Answer :

pieroraue

Answer:

We know that:

[tex]C_{(N)}=15000+8000N[/tex]

So first we need to substitute the next equation in the C(N) equation

[tex]N_{(t)} = 100t-5t^{2}[/tex]

And therefore we have:

[tex]C_{(t)}=15,000 + 8000*(100t-5t^{2})\\C_{(t)}=15,000 + 800000t-40000t^{2}[/tex]

Finally we replace at t = 5 in the equation above and we have:

[tex]C_{(5)}=15,000 + 800000*5-40000*5^{2}\\C_{(5)}= 3015000[/tex]

The interpretation is that after 5 hours of operation, we have wasted 3015000 dollars producing cars.  

The Interpretation of C(t) when t = 5 hours as a new function is;

In one day  after 5 hours, $3015000 would have been spent in production of cars.

  • We are given the function representing the cost 'C' (in dollars) of producing 'N' cars as;

C(N) = 15000 + 8000N

  • We are told that N' of cars produced at a certain factory in 1 day after 't' hours of operation is given by; N(t) = 100t - 5t²

Let us put 100t - 5t² for N in the cost equation to get;

C(t) = 15000 + 8000(100t - 5t²)

⇒ C(t) = 15000 + 800000t - 40000t²

We want to find C(t) when t = 5 hours. Thus;

C(5) = 15000 + 800000(5) - 40000(5²)

C(5) = $3015000

Thus, In one day  after 5 hours, $3015000 would have been spent in production of cars.

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