Assume it costs Microsoft $4,700 to manufacture 7 Xbox 360s and $8,690 to manufacture 14. Obtain the corresponding linear cost function.

To obtain the corresponding linear cost function, we need to determine the relationship between the number of Xbox 360s manufactured (x) and the cost to manufacture them (y).

We have two data points, (7, $4,700) and (14, $8,690).

To obtain the slope of the linear cost function, we can use the formula:

slope (m) = (change in y) / (change in x)

Let's calculate the slope using the two data points:

slope (m) = ($8,690 - $4,700) / (14 - 7)
= $3,990 / 7
= $570

Now that we have the slope, we can use the point-slope form of a linear equation to find the y-intercept (b). The point-slope form is:

y - y1 = m(x - x1)

We can choose either of the two data points. Let's choose (7, $4,700):

y - $4,700 = $570(x - 7)

Now, let's simplify the equation:

y - $4,700 = $570x - $3,990

To obtain the linear cost function in the form y = mx + b, we isolate y:

y = $570x - $3,990 + $4,700
= $570x + $710

Therefore, the corresponding linear cost function is:

C(x) = $570x + $710

To obtain the corresponding linear cost function, we can use the slope-intercept form of a linear equation, which is given by:

y = mx + b

where y represents the cost, x represents the number of Xbox 360s, m represents the slope (cost per Xbox 360), and b represents the y-intercept (fixed cost).

Let's find the slope first. We can use the formula for slope:

m = (y2 - y1) / (x2 - x1)

For the first set of data:
x1 = 7, y1 = 4700
x2 = 14, y2 = 8690

m = (8690 - 4700) / (14 - 7)
m = 3990 / 7
m = 570

Now, we can substitute one set of data to find the y-intercept (b). Let's use the first set of data:

y = mx + b
4700 = 570 * 7 + b
4700 = 3990 + b

b = 4700 - 3990
b = 710

Therefore, the corresponding linear cost function is:

y = 570x + 710

if x is the number of Xboxes,

x increases by 7, cost increases by 3990

so, the slope is 3990/7 = 570

cost = 570x + 710