Question

The function n(t)=−4t2+100t
represents the number of automobile parts produced at a factory after t hours. The function c(n)=20,000+5,000n
represents the cost in dollars of producing n car parts. What is (c∘n)(t)
and what does it mean? What is the domain?

Answer 1

To find the function \((c \circ n)(t)\), which represents the composition of the functions \(c(n)\) and \(n(t)\), we need to substitute the function \(n(t)\) into the function \(c(n)\).

The function \(n(t) = -4t^2 + 100t\) gives the number of automobile parts produced after \(t\) hours. The function \(c(n) = 20,000 + 5,000n\) gives the cost of producing \(n\) car parts.

### Step 1: Substitute \(n(t)\) into \(c(n)\)

\[
(c \circ n)(t) = c(n(t)) = c(-4t^2 + 100t)
\]

Now substitute \(-4t^2 + 100t\) into \(c(n)\):

\[
c(-4t^2 + 100t) = 20,000 + 5,000(-4t^2 + 100t)
\]

### Step 2: Simplify the expression

\[
c(-4t^2 + 100t) = 20,000 + 5,000(-4t^2) + 5,000(100t)
\]

This simplifies to:

\[
c(-4t^2 + 100t) = 20,000 - 20,000t^2 + 500,000t
\]

Combining the terms gives:

\[
(c \circ n)(t) = -20,000t^2 + 500,000t + 20,000
\]

### Meaning of \((c \circ n)(t)\)

The function \((c \circ n)(t)\) represents the total cost (in dollars) of producing automobile parts at the factory after \(t\) hours. In other words, it tells us how much it costs to produce parts as a function of time.

### Domain of \(n(t)\)

To find the domain of \((c \circ n)(t)\), we first need to consider the domain of \(n(t)\). Since \(n(t)\) is a quadratic function, we determine where \(n(t)\) is non-negative because the cost function \(c(n)\) is only defined for non-negative values of \(n\) (you can't produce a negative number of car parts).

Solving for when \(n(t) \geq 0\):

\[
-4t^2 + 100t \geq 0
\]

Factoring out \(t\):

\[
t(100 - 4t) \geq 0
\]

This gives us the critical points at \(t = 0\) and \(t = 25\) (where \(100 - 4t = 0\)).

Now, we can test intervals defined by these points:

1. For \(t < 0 \): \(n(t) < 0\) (not valid)
2. For \(0 < t < 25\): \(n(t) > 0\) (valid)
3. For \(t > 25\): \(n(t) < 0\) (not valid)

Thus, the valid values for \(t\) are in the interval:

\[
[0, 25]
\]

### Conclusion

So, the final result is:

\[
(c \circ n)(t) = -20,000t^2 + 500,000t + 20,000
\]
and the domain is

\[
t \in [0, 25].
\]