The total cost for a business to make greeting cards can be divided into one-time costs (e.g., a printing machine) and repeated costs (e.g., ink and paper). Suppose that the one-time cost to be able to make cards is $300, the total cost to make 300 cards is $900.00, and the total cost to make 650 cards is $1,600.00. What is the total cost to make 1,000 cards? Find the function rule to help you solve the problem.
(1 point)
Number of Cards Total Cost
0 $300
300 $900
650 $1,600
1,000
$2,100.00 $2,300.00 $2,500.00 $3,000.00
To find the function rule, we need to determine the relationship between the number of cards and the total cost. We can start by finding the difference in total cost and number of cards between two data points:
- To make 350 more cards (from 300 to 650), the total cost increased by $700 ($1600 - $900).
- This means that the cost per card for the repeated costs is $2 ($700 / 350).
Using this cost per card, we can calculate the total cost for 1000 cards:
- To make 350 more cards (from 650 to 1000), the total cost will increase by $700 ($2 x 350).
- Therefore, the total cost to make 1000 cards is $2300 ($1600 + $700).
So the function rule is:
Total cost = one-time cost + (cost per card x number of cards)
Plugging in the values we have:
Total cost = $300 + ($2 x number of cards)
Therefore, the total cost to make 1,000 cards is $2,300.00.
To find the total cost to make 1,000 cards, we need to establish a function rule that relates the number of cards to the total cost.
Let's first analyze the given information. We have three data points with corresponding numbers of cards and total costs:
- 300 cards cost $900.00
- 650 cards cost $1,600.00
To find the function rule, we can start by calculating the cost per card for each of these data points.
For 300 cards, the cost per card is $900.00 / 300 = $3.00 per card.
For 650 cards, the cost per card is $1,600.00 / 650 ≈ $2.46 per card.
This suggests that the cost per card decreases as the number of cards increases.
Now, to find the cost for 1,000 cards, we can use the cost per card from the data points.
For 1,000 cards, the estimated cost per card is $2.46 per card (we are assuming that the cost per card remains constant as the number of cards increases).
Therefore, the total cost to make 1,000 cards would be $2.46 x 1,000 = $2,460.00.
Using this information, we can update the table as follows:
Number of Cards Total Cost
0 $300
300 $900.00
650 $1,600.00
1,000 $2,460.00
To solve this problem, we need to find the relationship between the number of cards and the total cost, and then use that relationship to calculate the total cost for 1,000 cards.
Let's first examine the relationship between the number of cards and the total cost:
Number of Cards | Total Cost
0 | $300
300 | $900
650 | $1,600
We can see that as the number of cards increases, the total cost also increases. This indicates that the total cost is a linear function of the number of cards.
To find the function rule, we can use the linear equation:
Total Cost = m * Number of Cards + b
where "m" represents the slope and "b" represents the y-intercept.
We can use the given data points to find the slope:
m = (Total Cost2 - Total Cost1) / (Number of Cards2 - Number of Cards1)
= ($1,600 - $900) / (650 - 300)
= $700 / 350
= $2
By substituting one of the data points (e.g., (0, $300)) into the linear equation, we can solve for the y-intercept "b":
$300 = $2 * 0 + b
b = $300
Therefore, the function rule is:
Total Cost = $2 * Number of Cards + $300
Now, we can calculate the total cost to make 1,000 cards by substituting the number of cards into the function rule:
Total Cost = $2 * 1,000 + $300
Total Cost = $2,000 + $300
Total Cost = $2,300
Hence, the total cost to make 1,000 cards is $2,300.