Question

the function $d(p)$ gives the number of items that will be demanded when the price is $p$.
the production cost, $c(x)$, is the cost of producing $x$ items.

to determine the cost of production when the price is $\\$7$, you would:
\\( \circ \\) solve $c(d(p)) = 7$
\\( \circ \\) evaluate $d(c(7))$
\\( \circ \\) solve $d(c(x)) = 7$
\\( \circ \\) evaluate $c(d(7))$

Explanation:

1. First, understand the functions: \( D(p) \) gives the number of items demanded at price \( p \), and \( C(x) \) gives the cost of producing \( x \) items.
2. When the price is \( \$7 \), we first find the number of items demanded at this price, which is \( D(7) \) (since \( p = 7 \) in \( D(p) \)).
3. Then, we need to find the cost of producing that number of items. So we substitute \( x = D(7) \) into the cost function \( C(x) \), which means we evaluate \( C(D(7)) \).
4. Let's analyze the other options:

• "Solve \( C(D(p)) = 7 \)" is incorrect because we are not solving for \( p \) to make the cost 7, but finding the cost when price is 7.
• "Evaluate \( D(C(7)) \)" is incorrect because \( C(7) \) would be the cost of producing 7 items, and then \( D(C(7)) \) would be the demand at that cost, which is not what we want.
• "Solve \( D(C(x)) = 7 \)" is incorrect because we are not solving for \( x \) to make the demand at cost \( x \) equal to 7, but finding the cost when price is 7.

Answer:

D. Evaluate \( C(D(7)) \)