Question

The function p(t)=−6t2+500t
represents the number of widgets produced at a factory after t hours. The function c(w)=500+3.6w
represents the cost in dollars of producing w widgets. What is (c∘p)(t)
?

Enter the missing values.

(1 point)
$$
Remaining Attempts : 3
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Answer 1

To find \( (c \circ p)(t) \), we need to evaluate the function \( c(w) \) at \( w = p(t) \). Let's break this down step-by-step:

1. Start with the function \( p(t) = -6t^2 + 500t \), which gives us the number of widgets produced after \( t \) hours.
2. Next, we need to substitute this value into the function \( c(w) = 500 + 3.6w \).

Thus, we can calculate \( (c \circ p)(t) \):

\[
(c \circ p)(t) = c(p(t)) = c(-6t^2 + 500t)
\]

Now, substitute \( p(t) \) into \( c(w) \):

\[
c(-6t^2 + 500t) = 500 + 3.6(-6t^2 + 500t)
\]

Now distribute \( 3.6 \):

\[
= 500 + 3.6(-6t^2) + 3.6(500t)
\]
\[
= 500 - 21.6t^2 + 1800t
\]

Now combine the constants:

\[
= 2300 + 1800t - 21.6t^2
\]

So,

\[
(c \circ p)(t) = -21.6t^2 + 1800t + 2300
\]

In summary, \( (c \circ p)(t) = -21.6t^2 + 1800t + 2300 \).