IDNLearn.com offers a comprehensive solution for all your question and answer needs. Get the information you need from our community of experts who provide accurate and thorough answers to all your questions.

The table shows the total cost of purchasing the same priced items and a catalog.

[tex]\[
\begin{tabular}{|c|c|}
\hline
Number of Items & Total Cost \\
[tex]$( x )$[/tex] & [tex]$( y )$[/tex] \\
\hline
1 & \$10 \\
\hline
2 & \$14 \\
\hline
3 & \$18 \\
\hline
4 & \$22 \\
\hline
\end{tabular}
\][/tex]

What is the initial value and what does it represent?

A. \$4, the cost per item
B. \$4, the cost of the catalog
C. \$6, the cost per item
D. \$6, the cost of the catalog


Sagot :

To find the initial value and understand what it represents, let us analyze the given data step by step.

We are given the table of the total cost \( y \) for purchasing \( x \) same-priced items along with the catalog:

[tex]\[ \begin{array}{|c|c|} \hline \text{Number of Items } (x) & \text{Total Cost } (y) \\ \hline 1 & \$10 \\ \hline 2 & \$14 \\ \hline 3 & \$18 \\ \hline 4 & \$22 \\ \hline \end{array} \][/tex]

### Step-by-Step Solution:

1. Identify the relationship:
The total cost \( y \) is a linear function of the number of items \( x \). We can represent it in the form:
[tex]\[ y = mx + b \][/tex]
where:
- \( m \) is the slope (cost per item),
- \( b \) is the y-intercept (initial value, representing the cost of the catalog).

2. Calculate the slope \( m \):
The slope \( m \) can be calculated using any two points from the table. For instance, using the points \((1, 10)\) and \((2, 14)\):
[tex]\[ m = \frac{y_2 - y_1}{x_2 - x_1} \][/tex]
Substituting the values:
[tex]\[ m = \frac{14 - 10}{2 - 1} = \frac{4}{1} = 4 \][/tex]
Thus, the cost per item is \( \boldsymbol{\$4} \).

3. Find the y-intercept \( b \):
Using one of the points and the value of \( m \), we can solve for \( b \). Let's use the point \((1, 10)\):
[tex]\[ y = mx + b \][/tex]
Substituting \( y = 10 \), \( x = 1 \), and \( m = 4 \):
[tex]\[ 10 = 4 \cdot 1 + b \\ 10 = 4 + b \\ b = 10 - 4 = 6 \][/tex]
Therefore, the initial value \( b \) is \( \boldsymbol{\$6} \).

### Interpretation:
- \$4 represents the cost per item.
- \$6 represents the cost of the catalog.

So, the correct answers are:
- \(\$4\), the cost per item
- [tex]\(\$6\)[/tex], the cost of the catalog