Here’s the detailed step-by-step solution to the problem:
1. Understand the Components:
- Julie needs to cut 4 pieces of yarn, each with the same length [tex]\( x \)[/tex].
- She also needs to cut an additional piece of yarn that is 7.75 inches long.
2. Formulating the Equation:
- Let’s denote the length of each equal piece of yarn as [tex]\( x \)[/tex].
- Since she is cutting 4 pieces of yarn of length [tex]\( x \)[/tex], the total length of these equal pieces would be [tex]\( 4x \)[/tex].
- Additionally, she needs a piece that is 7.75 inches long.
- Therefore, the total length of all the pieces of yarn, [tex]\( y \)[/tex], would be the sum of the lengths of these equal pieces and the 7.75 inches long piece.
3. Writing the Equation:
- We sum the lengths of all the pieces to get the total length of yarn Julie cuts:
[tex]\[
y = 4x + 7.75
\][/tex]
4. Determining if the Graph is Continuous or Discrete:
- A graph is continuous if the variable [tex]\( x \)[/tex] can take any real number value within a given range. It would be discrete if [tex]\( x \)[/tex] can only take certain distinct values.
- In this context, [tex]\( x \)[/tex] represents the length of each piece of yarn, which can logically be any real number length within a practical range.
- Additionally, the total length [tex]\( y \)[/tex] would change continuously as [tex]\( x \)[/tex] changes continuously.
Therefore, the equation that can be used to determine the total length of yarn Julie ends up cutting is:
[tex]\[ y = 4x + 7.75 \][/tex]
The graph of this equation is continuous because [tex]\( x \)[/tex] can take any real number value.
Thus, the correct selection from the given options is:
[tex]\[ y = 4x + 7.75; \text{ continuous} \][/tex]
