Question
Jen just bought 120 feet of colorful ribbon. She is using the ribbon to make bows for teddy bears that she sells. Each bow uses 1.5 feet of ribbon. You can use a function to describe the number of feet of ribbon remaining after Jen makes x bows. Is the function linear or exponential? Which equation represents the function?
Answers
GPT-4o mini
The situation described involves Jen using a fixed amount of ribbon to make a number of bows. Each bow requires the same amount of ribbon (1.5 feet).
To determine whether the function is linear or exponential, we see that the remaining length of ribbon decreases by a consistent amount (1.5 feet) with each additional bow made. This indicates a linear relationship, rather than an exponential one (where the rate of change would vary).
We can define a function to describe the remaining length of ribbon after making \( x \) bows. Let's denote \( R(x) \) as the remaining ribbon in feet after making \( x \) bows. Initially, Jen has 120 feet of ribbon.
The equation can be derived as follows:
1. Start with the initial amount of ribbon: \( 120 \) feet.
2. Each bow uses \( 1.5 \) feet of ribbon, so after making \( x \) bows, the total amount of ribbon used is \( 1.5x \) feet.
3. Therefore, the amount of ribbon remaining can be expressed as:
\[
R(x) = 120 - 1.5x
\]
This function is linear since it can be represented in the standard form of a linear equation \( y = mx + b \), where:
- \( m \) (the slope) is \(-1.5\),
- \( b \) (the y-intercept) is \( 120 \).
In summary, the function is linear, and the equation that represents the function is:
\[
R(x) = 120 - 1.5x
\]
To determine whether the function is linear or exponential, we see that the remaining length of ribbon decreases by a consistent amount (1.5 feet) with each additional bow made. This indicates a linear relationship, rather than an exponential one (where the rate of change would vary).
We can define a function to describe the remaining length of ribbon after making \( x \) bows. Let's denote \( R(x) \) as the remaining ribbon in feet after making \( x \) bows. Initially, Jen has 120 feet of ribbon.
The equation can be derived as follows:
1. Start with the initial amount of ribbon: \( 120 \) feet.
2. Each bow uses \( 1.5 \) feet of ribbon, so after making \( x \) bows, the total amount of ribbon used is \( 1.5x \) feet.
3. Therefore, the amount of ribbon remaining can be expressed as:
\[
R(x) = 120 - 1.5x
\]
This function is linear since it can be represented in the standard form of a linear equation \( y = mx + b \), where:
- \( m \) (the slope) is \(-1.5\),
- \( b \) (the y-intercept) is \( 120 \).
In summary, the function is linear, and the equation that represents the function is:
\[
R(x) = 120 - 1.5x
\]