The number N of cars produced at a certain factory in 1 day after t hours of operation is given by Upper N (t) equals 800 t minus 5 t squared commaN (t) = 800t-5t2, 0 less than or equals t less than or equals 10.0≤t≤10. If the cost C (in dollars) of producing N cars is Upper C (Upper N) equals 30 comma 000 plus 8000 Upper N commaC (N) = 30,000+8000N, find the cost C as a function of the time of operation of the factory. What is the cost C as a function of the time t of operation of the factory?

The cost C as a function of t is C (t) = 30,000 + 6,400,000 t - 40,000 t²

Step-by-step explanation:

The function N (t) = 800 t - 5t², represents the number of cars produced at a time t hours in a day, where 0 ≤ t ≤ 10

The function C (N) = 30,000 + 8,000 N, represents the cost C (in dollars) of producing N cars

We need to find The cost C as a function of the time t

That means Substitute N in C by its function by other word find the composite function (C о N) (t)

∵ C (N) = 30,000 + 8,000 N

∵ N (t) = 800 t - 5 t²

- Substitute N in C by 800 t - 5 t²

∴ C (N (t)) = 30,000 + 8000 (800 t - 5 t²)

- Multiply the bracket by 8000

∴ C (N (t)) = 30,000 + 8000 (800 t) - 8000 (5 t²)

∴ C (N (t)) = 30,000 + 6,400,000 t - 40,000 t²

- C (N (t) = C (t)

∴ C (t) = 30,000 + 6,400,000 t - 40,000 t²

The cost C as a function of t is C (t) = 30,000 + 6,400,000 t - 40,000 t²