Question
Emily is organizing a tree-planting project at her school to celebrate Earth Day. She plans to order trees for delivery and wants to make sure her total cost, including a delivery fee and the cost of each tree, stays within her budget. There is a linear relationship between the number of trees Emily orders, and the total cost of the delivery. 2 4 6 8 10 12 14 16 18 20 $200 $400 $600 $800 $1,000 $1,200 $1,400 $1,600 $1,800 $2,000 x y Number of trees Total cost ($) Describe the rate of change for this relationship. The total cost of the delivery increases decreases by $1.50 $150.00 $167.00 $175.00 $200.00 $300.00 per tree.
Answers
Answered by
GPT-4o mini
To find the rate of change for the relationship between the number of trees ordered (x) and the total cost ($) (y), we need to analyze the data presented.
Based on the pairs of data provided and the linear relationship, we can calculate the rate of change (slope) using the formula:
\[ \text{Rate of Change} = \frac{\text{change in total cost}}{\text{change in number of trees}} \]
From the values given:
- When x = 2, y = $200
- When x = 4, y = $400
- When x = 6, y = $600
- ...
- Up to x = 20, y = $2000
We can see that as the number of trees (x) increases by 2 (for example, from 2 to 4), the total cost (y) increases by $200 (from $200 to $400).
Calculating the rate of change for these increments:
- Change in total cost when increasing from 2 to 4 trees:
\[ \text{Change in cost} = 400 - 200 = 200 \]
- Change in number of trees:
\[ \text{Change in trees} = 4 - 2 = 2 \]
So the rate of change (slope) is:
\[ \text{Rate of Change} = \frac{200}{2} = 100 \]
Now, calculating the total change per single tree:
If the cost increases by $200 for every 2 trees, it can be expressed to find the cost per one tree:
This gives us:
\[ \text{Cost per tree} = \frac{200}{2} = 100 \]
Thus, the total cost increases by $100.00 per tree.
Based on the options provided:
- **The total cost of the delivery increases by $100.00 per tree.**
Therefore, your conclusion for the rate of change in this relationship is that the total cost increases by **$100.00** for each additional tree ordered.
Based on the pairs of data provided and the linear relationship, we can calculate the rate of change (slope) using the formula:
\[ \text{Rate of Change} = \frac{\text{change in total cost}}{\text{change in number of trees}} \]
From the values given:
- When x = 2, y = $200
- When x = 4, y = $400
- When x = 6, y = $600
- ...
- Up to x = 20, y = $2000
We can see that as the number of trees (x) increases by 2 (for example, from 2 to 4), the total cost (y) increases by $200 (from $200 to $400).
Calculating the rate of change for these increments:
- Change in total cost when increasing from 2 to 4 trees:
\[ \text{Change in cost} = 400 - 200 = 200 \]
- Change in number of trees:
\[ \text{Change in trees} = 4 - 2 = 2 \]
So the rate of change (slope) is:
\[ \text{Rate of Change} = \frac{200}{2} = 100 \]
Now, calculating the total change per single tree:
If the cost increases by $200 for every 2 trees, it can be expressed to find the cost per one tree:
This gives us:
\[ \text{Cost per tree} = \frac{200}{2} = 100 \]
Thus, the total cost increases by $100.00 per tree.
Based on the options provided:
- **The total cost of the delivery increases by $100.00 per tree.**
Therefore, your conclusion for the rate of change in this relationship is that the total cost increases by **$100.00** for each additional tree ordered.