Question
A graph with x-axis and y-axis labeled from 0 to 30 in increments of 2. Three points are plotted with a line connecting them.
A graph with x-axis and y-axis labeled from 0 to 30 in increments of 2. Points are plotted at left parenthesis 2 comma 10 right parenthesis, left parenthesis 3 comma 15 right parenthesis, and left parenthesis 6 comma 30 right parenthesis. A line is drawn connecting the three points.
The stretch of a spring in millimeters is proportional to the weight in grams attached to the end of the spring. Find the unit rate.
A graph with x-axis and y-axis labeled from 0 to 30 in increments of 2. Points are plotted at left parenthesis 2 comma 10 right parenthesis, left parenthesis 3 comma 15 right parenthesis, and left parenthesis 6 comma 30 right parenthesis. A line is drawn connecting the three points.
The stretch of a spring in millimeters is proportional to the weight in grams attached to the end of the spring. Find the unit rate.
Answers
GPT-4o mini
To find the unit rate of the stretch of a spring in relation to the weight attached, we can examine the points provided: (2, 10), (3, 15), and (6, 30).
The unit rate describes how much the stretch of the spring (in millimeters) increases for each gram of weight.
We can use any of the points to find the unit rate. It is often simplest to use the slope between two points to determine this. The formula for the slope (or unit rate) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
\]
Let's calculate the unit rate using the first two points \((2, 10)\) and \((3, 15)\).
Using \((x_1, y_1) = (2, 10)\) and \((x_2, y_2) = (3, 15)\):
\[
\text{slope} = \frac{15 - 10}{3 - 2} = \frac{5}{1} = 5
\]
This means that for every gram of weight added to the spring, the stretch increases by 5 millimeters.
To ensure consistency, we can also check the slope between the second and third points \((3, 15)\) and \((6, 30)\):
Using \((x_1, y_1) = (3, 15)\) and \((x_2, y_2) = (6, 30)\):
\[
\text{slope} = \frac{30 - 15}{6 - 3} = \frac{15}{3} = 5
\]
Again, we find that the unit rate is 5 millimeters of stretch per gram of weight.
Thus, the unit rate is **5 mm/g** (millimeters per gram).
The unit rate describes how much the stretch of the spring (in millimeters) increases for each gram of weight.
We can use any of the points to find the unit rate. It is often simplest to use the slope between two points to determine this. The formula for the slope (or unit rate) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by:
\[
\text{slope} = \frac{y_2 - y_1}{x_2 - x_1}
\]
Let's calculate the unit rate using the first two points \((2, 10)\) and \((3, 15)\).
Using \((x_1, y_1) = (2, 10)\) and \((x_2, y_2) = (3, 15)\):
\[
\text{slope} = \frac{15 - 10}{3 - 2} = \frac{5}{1} = 5
\]
This means that for every gram of weight added to the spring, the stretch increases by 5 millimeters.
To ensure consistency, we can also check the slope between the second and third points \((3, 15)\) and \((6, 30)\):
Using \((x_1, y_1) = (3, 15)\) and \((x_2, y_2) = (6, 30)\):
\[
\text{slope} = \frac{30 - 15}{6 - 3} = \frac{15}{3} = 5
\]
Again, we find that the unit rate is 5 millimeters of stretch per gram of weight.
Thus, the unit rate is **5 mm/g** (millimeters per gram).