Question

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 1

Answer:

We know that:

$C_{(N)}=15000+8000N$

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

$N_{(t)} = 100t-5t^{2}$

And therefore we have:

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

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

$C_{(5)}=15,000 + 800000*5-40000*5^{2}\\C_{(5)}= 3015000$

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

Answer 2

Given: C(N) = 15,000 + 8000N 

In the above equation simply substitute: 
N(t) = 100t - 5t^2 
for N 

Therefore:
C(t) = 15,000 + 8000{ 100t-5t^2 } 
C(t) =15,000 + 800,000t - 40,000t^2.

 

at t = 5

C(5) = 15,000 + 800,000*5 - 40,000*(5)^2

C(5) = 3,015,000