Question

A company finds it can produce 5 heaters for $1750, while producing 10 heaters cost $3000. Express the cost, y, as a linear function of the number of heaters, x.

Answer 1

The cost as a function of the number of heaters is represented by the linear equation y = $250x + $500, where the slope is $250 representing the cost per additional heater and the y-intercept is $500 representing the fixed costs.

Step-by-step explanation:

To express the cost y as a linear function of the number of heaters x, we can use two given points to determine the slope of the line and then use one of the points to find the y-intercept. The two points we have from the information provided are (5, $1750) and (10, $3000).

First, we calculate the slope (m), which is the change in cost divided by the change in the number of heaters.

m = ($3000 - $1750) / (10 - 5) = $1250 / 5 = $250 per heater

Now that we have the slope, we can use point-slope form to determine the y-intercept. We'll use the point (5, $1750).

$1750 = $250(5) + b

To find b, we solve for y-intercept:

b = $1750 - $250(5) = $1750 - $1250 = $500

The linear equation representing the cost based on the number of heaters is:

y = $250x + $500

Answer 2

The cost function is linear, and as such has a slope and a y-intercept.

Two points on this line are (5, $1750) and (10, $3000). As the number of heaters produced increases from 5 to 10, the production costs increase by $1250. Thus, the slope of this line is m = $1250/5, or m = $250/unit.

Inserting known data into the slope-intercept form of the equation of a straight line, we get:

$3000 = ($250/unit)(10) + b. Then $500 = b, and the cost function is:

y = ($250/unit)(x) + $500.