The question is an illustration of composite functions.
- Functions c(n) and h(n) are
and ![\mathbf{n(h) = 5h}](https://tex.z-dn.net/?f=%5Cmathbf%7Bn%28h%29%20%3D%205h%7D)
- The composite function c(n(h)) is
![\mathbf{c(n(h)) = 5000 + 1250h}](https://tex.z-dn.net/?f=%5Cmathbf%7Bc%28n%28h%29%29%20%3D%205000%20%2B%201250h%7D)
- The value of c(n(100)) is
![\mathbf{c(n(100)) = 130000}](https://tex.z-dn.net/?f=%5Cmathbf%7Bc%28n%28100%29%29%20%3D%20130000%7D)
- The interpretation is: <em>"the cost of working for 100 hours is $130000"</em>
The given parameters are:
- $5000 in fixed costs plus an additional $250
- 5 systems in one hour of production
<u>(a) Functions c(n) and n(h)</u>
Let the number of system be n, and h be the number of hours
So, the cost function (c(n)) is:
![\mathbf{c(n) = Fixed + Additional \times n}](https://tex.z-dn.net/?f=%5Cmathbf%7Bc%28n%29%20%3D%20Fixed%20%2B%20Additional%20%5Ctimes%20n%7D)
This gives
![\mathbf{c(n) = 5000 + 250 \times n}](https://tex.z-dn.net/?f=%5Cmathbf%7Bc%28n%29%20%3D%205000%20%2B%20250%20%5Ctimes%20n%7D)
![\mathbf{c(n) = 5000 + 250n}](https://tex.z-dn.net/?f=%5Cmathbf%7Bc%28n%29%20%3D%205000%20%2B%20250n%7D)
The function for number of systems is:
![\mathbf{n(h) = 5 \times h}](https://tex.z-dn.net/?f=%5Cmathbf%7Bn%28h%29%20%3D%205%20%5Ctimes%20h%7D)
![\mathbf{n(h) = 5h}](https://tex.z-dn.net/?f=%5Cmathbf%7Bn%28h%29%20%3D%205h%7D)
<u>(b) Function c(n(h))</u>
In (a), we have:
![\mathbf{c(n) = 5000 + 250n}](https://tex.z-dn.net/?f=%5Cmathbf%7Bc%28n%29%20%3D%205000%20%2B%20250n%7D)
![\mathbf{n(h) = 5h}](https://tex.z-dn.net/?f=%5Cmathbf%7Bn%28h%29%20%3D%205h%7D)
Substitute n(h) for n in ![\mathbf{c(n) = 5000 + 250n}](https://tex.z-dn.net/?f=%5Cmathbf%7Bc%28n%29%20%3D%205000%20%2B%20250n%7D)
![\mathbf{c(n(h)) = 5000 + 250n(h)}](https://tex.z-dn.net/?f=%5Cmathbf%7Bc%28n%28h%29%29%20%3D%205000%20%2B%20250n%28h%29%7D)
Substitute ![\mathbf{n(h) = 5h}](https://tex.z-dn.net/?f=%5Cmathbf%7Bn%28h%29%20%3D%205h%7D)
![\mathbf{c(n(h)) = 5000 + 250 \times 5h}](https://tex.z-dn.net/?f=%5Cmathbf%7Bc%28n%28h%29%29%20%3D%205000%20%2B%20250%20%5Ctimes%205h%7D)
![\mathbf{c(n(h)) = 5000 + 1250h}](https://tex.z-dn.net/?f=%5Cmathbf%7Bc%28n%28h%29%29%20%3D%205000%20%2B%201250h%7D)
<u>(c) Find c(n(100))</u>
c(n(100)) means that h = 100.
So, we have:
![\mathbf{c(n(100)) = 5000 + 1250 \times 100}](https://tex.z-dn.net/?f=%5Cmathbf%7Bc%28n%28100%29%29%20%3D%205000%20%2B%201250%20%5Ctimes%20100%7D)
![\mathbf{c(n(100)) = 5000 + 125000}](https://tex.z-dn.net/?f=%5Cmathbf%7Bc%28n%28100%29%29%20%3D%205000%20%2B%20125000%7D)
![\mathbf{c(n(100)) = 130000}](https://tex.z-dn.net/?f=%5Cmathbf%7Bc%28n%28100%29%29%20%3D%20130000%7D)
<u>(d) Interpret (c)</u>
In (c), we have: ![\mathbf{c(n(100)) = 130000}](https://tex.z-dn.net/?f=%5Cmathbf%7Bc%28n%28100%29%29%20%3D%20130000%7D)
It means that:
The cost of working for 100 hours is $130000
Read more about composite functions at:
brainly.com/question/10830110