Question

A company finds it can produce 15 heaters for $ 4000​, while producing 20 heaters costs $ 5200. Express the​ cost, y, as a linear function of the number of​ heaters, x.
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Part 1
Express the​ cost, y, as a linear function of the number of​ heaters, x.
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Answer 1

To express the cost \( y \) as a linear function of the number of heaters \( x \), we can use the two points given in the problem. The points are:

1. \( (15, 4000) \) — when producing 15 heaters, the cost is $4000.
2. \( (20, 5200) \) — when producing 20 heaters, the cost is $5200.

We will first determine the slope \( m \) of the line that represents the cost function. The slope formula between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is:

\[
m = \frac{y_2 - y_1}{x_2 - x_1}
\]

Substituting the values from our points:

\[
m = \frac{5200 - 4000}{20 - 15} = \frac{1200}{5} = 240
\]

So, the slope of the line is \( 240 \).

Next, we can use the slope-intercept formula for a line, which is:

\[
y = mx + b
\]

We already have the slope \( m = 240 \). Now we need to find the y-intercept \( b \). We can use one of the points to solve for \( b \). Let's use the point \( (15, 4000) \):

\[
4000 = 240(15) + b
\]

Calculating \( 240(15) \):

\[
240 \times 15 = 3600
\]

Now substituting back into the equation:

\[
4000 = 3600 + b
\]
\[
b = 4000 - 3600 = 400
\]

Now we have both the slope and y-intercept:

- Slope \( m = 240 \)
- Y-intercept \( b = 400 \)

Thus, the cost \( y \) as a linear function of the number of heaters \( x \) is:

\[
y = 240x + 400
\]

So, the final answer is:

\[
\boxed{y = 240x + 400}
\]