SOLUTION: A company has a manufacturing plant that is producing quality lamps. They find that in order to produce 220 lamps in a month, it will cost $9180. Also, to produce 460 lamps in a mo

Question 1149987: A company has a manufacturing plant that is producing quality lamps. They find that in order to produce 220 lamps in a month, it will cost $9180. Also, to produce 460 lamps in a month, it will cost $16140. Find an equation in the form y=mx+b,where x is the number of lamps produced in a month and y is the monthly cost to do so.

Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!

x = number of lamps produced in a month
y = monthly cost to make x lamps

Fact 1: in order to produce 220 lamps in a month, it will cost $9180.
Fact 2: to produce 460 lamps in a month, it will cost $16140

From fact 1, we know that (x,y) = (220, 9180)
From fact 2, we know that (x,y) = (460, 16140)

Let,
(x1,y1) = (220, 9180)
(x2,y2) = (460, 16140)
Use the slope formula
m = (y2-y1)/(x2-x1)
m = (16140-9180)/(460-220)
m = 6960/240
m = 29

The slope is 29 = 29/1
Slope = rise/run = 29/1
rise = 29 = change in y
run = 1 = change in x
Each time x goes up by 1, the value of y goes up by 29
In this context, it means that each lamp costs $29 to make. Or you can say the unit cost is 29 dollars per lamp.

We'll use m = 29 along with one of the given points. Let's use the point (x,y) = (220,9180)
Plug these three items into y = mx+b and isolate b.

y = mx+b
9180 = 29*220+b
9180 = 6380+b
6380+b = 9180
b+6380 = 9180
b+6380-6380 = 9180-6380 ... subtract 6380 from both sides
b = 2800

The y intercept is 2800, meaning the graph crosses the y axis at that location.
The ordered pair of this y intercept is (0,2800)
Interpretation: If you make 0 lamps, then you still have $2800 in costs.
This is the fixed cost (eg: rent).

With m = 29 and b = 2800 we go from y = mx+b to y = 29x + 2800

------------------------------------

Let's see what happens when we plug x = 220 into the equation
y = 29x + 2800
y = 29*220 + 2800
y = 6380 + 2800
y = 9180
As expected, x = 29 lamps cost a total of y = 9180 dollars.

Now let's see what happens when we plug in x = 460
y = 29x + 2800
y = 29*460 + 2800
y = 13340 + 2800
y = 16140
So x = 460 lamps cost a total of y = 16140 dollars, which matches with what was given.

Both of the results in this section help confirm we have the right equation.

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