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:

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