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

To calculate the cost of production as a function of time, substitute the production function N(t) into the cost formula C(N). After simplifying, the cost function is C(t) = 15,000 + 800,000t - 40,000t². At 5 hours, the cost is 3,015,000 dollars.

Step-by-step explanation:

To find the cost C as a function of the time t of operation of the factory, we first use the formula for the number of cars produced N(t) = 100t - 5t². Then we substitute the expression for N(t) into the cost function C(N) = 15,000 + 8000N.

Substituting N(t) into C(N), we get:

1. C(N(t)) = 15,000 + 8000(100t - 5t²)
2. C(t) = 15,000 + 8000(100t) - 8000(5t²)
3. C(t) = 15,000 + 800,000t - 40,000t²

This is the cost C as a function of time t.

When t = 5 hours, the function becomes:

• C(5) = 15,000 + 800,000(5) - 40,000(5²)
• C(5) = 15,000 + 4,000,000 - 1,000,000
• C(5) = 3,015,000 dollars

This result means that the cost of production after 5 hours of operation is 3,015,000 dollars.

Answer 2

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.