To determine the most logical interpretation of the equation [tex]\(C(20) = 700\)[/tex] given the cost function [tex]\(C(L) = 300 + 20L\)[/tex], let's go through the steps to understand what this function represents and how the given values fit into it.
1. Understand the given cost function:
[tex]\[
C(L) = 300 + 20L
\][/tex]
This equation states that the cost [tex]\(C\)[/tex] for installing a sewer pipe consists of a fixed cost of [tex]$300 plus an additional $[/tex]20 for each foot of the pipe. Here,
- [tex]\(300\)[/tex] is the fixed cost.
- [tex]\(20L\)[/tex] is the variable cost that depends on the length [tex]\(L\)[/tex] of the pipe in feet.
2. Interpret the given value:
- The equation [tex]\(C(20) = 700\)[/tex] involves substituting [tex]\(L = 20\)[/tex] feet into the cost function.
3. Substitute [tex]\(L = 20\)[/tex] into the cost function:
[tex]\[
C(20) = 300 + 20 \times 20
\][/tex]
4. Calculate the cost:
[tex]\[
C(20) = 300 + 400 = 700
\][/tex]
This calculation confirms that for [tex]\(L = 20\)[/tex] feet, the cost is $700.
5. Interpret the calculated result:
The calculated cost [tex]\(700\)[/tex] corresponds to installing a pipe that is [tex]\(20\)[/tex] feet long.
Given the options, let's examine which matches our interpretation:
- (A) The cost to install a 700-foot sewer pipe is 20 dollars.
- This is incorrect because it misinterprets both the cost and the length in the context of the cost function.
- (B) 20 times the cost of a typical pipe is equivalent to 700 dollars.
- This is incorrect because it does not directly relate to the cost function given or to the interpretation of the calculated result.
- (C) The cost to install a 20-foot-long sewer pipe is 700 dollars.
- This is correct because it directly matches the interpretation of our calculated result.
Therefore, the most logical interpretation of the equation [tex]\(C(20) = 700\)[/tex] is:
[tex]\[
\boxed{\text{(C) The cost to install a 20-foot-long sewer pipe is 700 dollars.}}
\][/tex]
