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Each week, Heather’s company has $5000 in fixed costs plus an additional $250 for each system produced. The company is able to produce 5 systems in one hour of production, and h represent the number of hours in production

Part a] write functions c(n) and n(h) to model this situation, explain what they represent

Part b] Then write a function c(n(h)) to represent the coat incurred in h hours. Show the work or explain the reasoning used to determine the answer

Part c] Find c(n(100)).

Part d] interpret your solution to part c

Respuesta :

MrRoyal

The question is an illustration of composite functions.

  • Functions c(n) and h(n) are [tex]\mathbf{c(n) = 5000 + 250n}[/tex] and [tex]\mathbf{n(h) = 5h}[/tex]
  • The composite function c(n(h)) is [tex]\mathbf{c(n(h)) = 5000 + 1250h}[/tex]
  • The value of c(n(100)) is [tex]\mathbf{c(n(100)) = 130000}[/tex]
  • The interpretation is: "the cost of working for 100 hours is $130000"

The given parameters are:

  • $5000 in fixed costs plus an additional $250
  • 5 systems in one hour of production

(a) Functions c(n) and n(h)

Let the number of system be n, and h be the number of hours

So, the cost function (c(n)) is:

[tex]\mathbf{c(n) = Fixed + Additional \times n}[/tex]

This gives

[tex]\mathbf{c(n) = 5000 + 250 \times n}[/tex]

[tex]\mathbf{c(n) = 5000 + 250n}[/tex]

The function for number of systems is:

[tex]\mathbf{n(h) = 5 \times h}[/tex]

[tex]\mathbf{n(h) = 5h}[/tex]

(b) Function c(n(h))

In (a), we have:

[tex]\mathbf{c(n) = 5000 + 250n}[/tex]

[tex]\mathbf{n(h) = 5h}[/tex]

Substitute n(h) for n in [tex]\mathbf{c(n) = 5000 + 250n}[/tex]

[tex]\mathbf{c(n(h)) = 5000 + 250n(h)}[/tex]

Substitute [tex]\mathbf{n(h) = 5h}[/tex]

[tex]\mathbf{c(n(h)) = 5000 + 250 \times 5h}[/tex]

[tex]\mathbf{c(n(h)) = 5000 + 1250h}[/tex]

(c) Find c(n(100))

c(n(100)) means that h = 100.

So, we have:

[tex]\mathbf{c(n(100)) = 5000 + 1250 \times 100}[/tex]

[tex]\mathbf{c(n(100)) = 5000 + 125000}[/tex]

[tex]\mathbf{c(n(100)) = 130000}[/tex]

(d) Interpret (c)

In (c), we have: [tex]\mathbf{c(n(100)) = 130000}[/tex]

It means that:

The cost of working for 100 hours is $130000

Read more about composite functions at:

https://brainly.com/question/10830110

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