A company finds it can produce 25 heaters for $ 6100, while producing 30 heaters costs $ 7200. Express the cost, y, as a linear function of the number of heaters, x.
y=
5 additional heaters cost $1100 ... $220 per heater ... this is the slope
y = 220 x + b
plug in one of the points to find b
To find the linear function for the cost in terms of the number of heaters, we can use the slope-intercept form of a linear equation: y = mx + b, where m is the slope and b is the y-intercept.
First, let's find the slope (m) using the two given data points:
Point 1: (25, $6100)
Point 2: (30, $7200)
The slope (m) can be calculated as:
m = (change in y) / (change in x)
= (7200 - 6100) / (30 - 25)
= 1100 / 5
= 220
Now, using one of the points (25, $6100) and the slope (m = 220), we can find the y-intercept (b) by substituting the values into the slope-intercept form:
y = mx + b
$6100 = 220 * 25 + b
$6100 = 5500 + b
To solve for b, we can subtract 5500 from both sides:
$6100 - $5500 = b
$600 = b
Therefore, the linear function for the cost (y) in terms of the number of heaters (x) is:
y = 220x + $600
To express the cost, y, as a linear function of the number of heaters, x, we need to find the equation of a line that relates the cost to the number of heaters.
Given that the company can produce 25 heaters for $6100 and 30 heaters for $7200, we have two data points: (25, 6100) and (30, 7200).
First, we need to find the slope of the line. The slope, denoted as m, is calculated using the formula:
m = (y2 - y1) / (x2 - x1)
Using the points (25, 6100) and (30, 7200), the slope is:
m = (7200 - 6100) / (30 - 25)
= 1100 / 5
= 220
Now that we have the slope, we can use the point-slope form of a linear equation:
y - y1 = m(x - x1)
Using the point (25, 6100):
y - 6100 = 220(x - 25)
To simplify, let's distribute the 220:
y - 6100 = 220x - 5500
Now, let's isolate y:
y = 220x - 5500 + 6100
y = 220x + 600
The linear function that expresses the cost, y, as a function of the number of heaters, x, is:
y = 220x + 600