Julie needs to cut 4 pieces of yarn, each with the same length, and a piece of yarn 7.75 inches long. Let [tex]$x$[/tex] represent the length of each of the equal pieces of yarn that Julie decides to cut. What is the equation that can be used to determine the total length of all of the yarn that she ends up cutting, [tex]$y$[/tex]? Is the graph of the equation continuous or discrete?
A. [tex]$y=7.75 x+4$[/tex]; discrete
B. [tex]$y=7.75 x+4$[/tex]; continuous
C. [tex]$y=4 x+7.75$[/tex]; discrete
D. [tex]$y=4 x+7.75$[/tex]; continuous
Answer (1)
The total length of the 4 equal pieces of yarn is 4 x .
The total length of all yarn is the sum of the length of the 4 equal pieces and the additional piece: y = 4 x + 7.75 .
The length x can be any non-negative real number.
The graph of the equation is continuous, so the final answer is y = 4 x + 7.75 ; continuous .
Explanation
Problem Analysis Let's analyze the problem. Julie cuts 4 pieces of yarn of equal length, and one additional piece of yarn with a fixed length of 7.75 inches. We need to find an equation that represents the total length of the yarn, y , in terms of the length of each of the equal pieces, x . We also need to determine if the graph of this equation is continuous or discrete.
Equation Derivation The total length of the 4 equal pieces of yarn is 4 x inches, since each piece has a length of x inches. The total length of all the yarn is the sum of the length of the 4 equal pieces and the additional piece, which is 4 x + 7.75 inches. Therefore, the equation that represents the total length of the yarn is y = 4 x + 7.75 .
Continuity Analysis Now, let's determine if the graph of the equation is continuous or discrete. The length x can take on any non-negative real value, as Julie can cut the yarn to any length (within practical limits). Since x can be any non-negative real number, the graph of the equation y = 4 x + 7.75 is continuous.
Final Answer Therefore, the correct equation is y = 4 x + 7.75 , and the graph of the equation is continuous.
Examples
Imagine you're making friendship bracelets. You need four strands of the same length for each bracelet, plus an extra 7.75 inches for the clasp and tying. The equation y = 4 x + 7.75 helps you calculate the total length of string you need for each bracelet, where x is the length of each of the four equal strands. This kind of linear equation is useful in many situations where you have a fixed amount plus a variable amount that depends on a certain quantity, such as calculating costs, distances, or material requirements.
