In Beloy’s fish ball factory, it costs 25 to make 15 pieces of fishball while it costs 40 to make 27 pieces.  Assuming that the total cost is described by a linear function, how much does it cost to make 19 pieces of fishball

To solve this problem, we can assume that the total cost of making fishballs is described by a linear function of the number of pieces. Let's denote the cost by \( C \) and the number of pieces by \( n \). The linear function can be expressed as:

\[
C = mn + b
\]

where \( m \) is the slope (cost per piece), and \( b \) is the fixed cost.

### Step 1: Set up the equations

We have two data points from the problem:
1. \( C(15) = 25 \)
2. \( C(27) = 40 \)

These give us the following system of linear equations:
\[
25 = 15m + b \quad \text{(1)}
\]
\[
40 = 27m + b \quad \text{(2)}
\]

### Step 2: Subtract the first equation from the second to find \( m \)
Subtract equation (1) from equation (2) to eliminate \( b \):
\[
(40 - 25) = (27m + b) - (15m + b)
\]
\[
15 = 12m
\]
\[
m = \frac{15}{12} = \frac{5}{4} = 1.25
\]

So, the slope \( m = 1.25 \).

### Step 3: Solve for \( b \) using one of the original equations
Now, substitute \( m = 1.25 \) into equation (1) to solve for \( b \):
\[
25 = 15(1.25) + b
\]
\[
25 = 18.75 + b
\]
\[
b = 25 - 18.75 = 6.25
\]

So, the fixed cost \( b = 6.25 \).

### Step 4: Find the cost for 19 pieces
Now, we can use the linear function to find the cost \( C \) for 19 pieces:
\[
C(19) = 1.25(19) + 6.25
\]
\[
C(19) = 23.75 + 6.25 = 30
\]

### Final Answer:
It costs **30** to make 19 pieces of fishball.

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Do you want more details or have any questions? Here are some related questions:

1. How would the cost change if the number of pieces increased to 35?
2. What is the cost per fishball?
3. What would the fixed cost be if it were doubled?
4. How does the slope \( m \) affect the overall cost?
5. What would be the cost function if the cost for 10 pieces was known?
6. How would the total cost change if the price per piece increased by 10%?
7. What is the break-even point if the price per fishball is 2?
8. Can the linear model be used for very large or very small numbers of fishballs?

**Tip:** Always check your calculations by substituting back into the original equations to ensure consistency.

Explore how to calculate the cost of making fishballs using linear functions. Solve for the cost to make 19 fishballs based on given production costs for 15 and 27 fishballs. Suitable for Grades 7-9.

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