Question
Abby purchased a new home and is eager to create a lush, green lawn. To achieve this, she bought a large bag of grass seed to reseed her yard. First, she will spread the seed evenly over the lawn, and then she will set up a sprinkler to water it.
There is a linear relationship between the lawn area Abby has covered with grass seed, in square feet, and the amount of seed left in the bag, in pounds.
There is a linear relationship between the lawn area Abby has covered with grass seed, in square feet, and the amount of seed left in the bag, in pounds.
Answers
GPT-4o mini
It sounds like Abby is embarking on a great project to establish a beautiful lawn! To better understand the linear relationship between the lawn area covered with grass seed and the amount of seed remaining in the bag, we can explore some key concepts.
1. **Understanding Linear Relationships**: In a linear relationship, one variable changes at a constant rate with respect to another variable. This can be expressed with the equation of a line in slope-intercept form:
\[
y = mx + b
\]
where:
- \( y \) represents the amount of seed left in pounds,
- \( x \) represents the area covered in square feet,
- \( m \) is the slope (the rate at which the seed is used per square foot), and
- \( b \) is the y-intercept (the initial amount of seed before any is used).
2. **Setting Up the Relationship**: Let’s assume Abby starts with \( S \) pounds of seed. If she uses \( r \) pounds of seed per square foot of lawn, the relationship can be expressed as:
\[
\text{Amount of seed left (in pounds)} = S - r \times \text{Area (in square feet)}
\]
3. **Finding Key Values**:
- **Initial Seed Amount**: \( S \) is the total weight of the seed in the bag.
- **Seed Usage Rate**: \( r \) is determined by how much seed she uses per square foot covered.
- **Current Area Covered**: As Abby spreads the seed, the area \( x \) increases, and the unused seed \( y \) decreases.
4. **Example**: If Abby starts with 50 pounds of seed and uses 0.05 pounds per square foot, the equation becomes:
\[
y = 50 - 0.05x
\]
Here, if she covers 1000 square feet, she would have:
\[
y = 50 - 0.05 \times 1000 = 50 - 50 = 0 \text{ pounds of seed left}
\]
This example illustrates how the amount of seed decreases as the area increases, emphasizing the linear relationship.
5. **Graphing the Relationship**: If plotted on a graph, the y-axis could represent the amount of seed left (in pounds), and the x-axis could represent the area covered (in square feet). The line would start at \( S \) on the y-axis and decrease until it reaches the x-axis where all the seed is used up.
Abby can monitor her progress by using this relationship. As she covers more of her lawn, she can calculate how much seed she has left using this linear equation!
1. **Understanding Linear Relationships**: In a linear relationship, one variable changes at a constant rate with respect to another variable. This can be expressed with the equation of a line in slope-intercept form:
\[
y = mx + b
\]
where:
- \( y \) represents the amount of seed left in pounds,
- \( x \) represents the area covered in square feet,
- \( m \) is the slope (the rate at which the seed is used per square foot), and
- \( b \) is the y-intercept (the initial amount of seed before any is used).
2. **Setting Up the Relationship**: Let’s assume Abby starts with \( S \) pounds of seed. If she uses \( r \) pounds of seed per square foot of lawn, the relationship can be expressed as:
\[
\text{Amount of seed left (in pounds)} = S - r \times \text{Area (in square feet)}
\]
3. **Finding Key Values**:
- **Initial Seed Amount**: \( S \) is the total weight of the seed in the bag.
- **Seed Usage Rate**: \( r \) is determined by how much seed she uses per square foot covered.
- **Current Area Covered**: As Abby spreads the seed, the area \( x \) increases, and the unused seed \( y \) decreases.
4. **Example**: If Abby starts with 50 pounds of seed and uses 0.05 pounds per square foot, the equation becomes:
\[
y = 50 - 0.05x
\]
Here, if she covers 1000 square feet, she would have:
\[
y = 50 - 0.05 \times 1000 = 50 - 50 = 0 \text{ pounds of seed left}
\]
This example illustrates how the amount of seed decreases as the area increases, emphasizing the linear relationship.
5. **Graphing the Relationship**: If plotted on a graph, the y-axis could represent the amount of seed left (in pounds), and the x-axis could represent the area covered (in square feet). The line would start at \( S \) on the y-axis and decrease until it reaches the x-axis where all the seed is used up.
Abby can monitor her progress by using this relationship. As she covers more of her lawn, she can calculate how much seed she has left using this linear equation!