y = 10 * x
y = 80000 + 8 * x
y = 120000 + 120000 + 5 * x
y is the total costs.
x is the number of units.
if you have a graphing calculator, such as he one at https://www.desmos.com/calculator, you can get a visual idea of what the answer will be, as shown below:
you can see that y = 10x is the minimum cost solution from 0 < x < 24000 and y = 120000 + 5x from 24000 < x < infinity.
to find the answer algebraically, i think the best way is to find the breaking points and then look at the intervals in between.
first compare y = 10x to y = 80,000 + 8x to see at what point they break even.
this happens when 10x = 80,000 + 8x.
subtract 8x from both sides of the equation to get:
2x = 80,000.
solve for x to get x = 40,000.
next compare y = 10x to y = 120,000 + 5x to see at what point they break even.
this happens when 10x = 120,000 + 5x.
subtract 5x from both sides of the equation to get:
5x = 120,000
solve for x to get x = 24,000.
next compare y = 80,000 + 8x to y = 120,000 + 5x to see at what point they break even.
this happens when 80,000 + 8x = 120,000 + 5x
subtract 5x from both sides of the equation and subtract 80,000 from both sides of the equation to get:
3x = 40,000
solve for x to get x = 13,333 and 1/3.
you have 3 breaking points.
they are:
x = 40,000
x = 24,000
x = 13,333 and 1/3
your intervals will be:
0 < x < 13,333 and 1/3
13,333 and 1/3 < x < 24,000
24,000 < x < 40,000
40,000 < x < infinity.
you can pick any value within these intervals to see which is the minimum cost.
i usually pick a value that's easy to calculate within each interval.
at x = 10,000, which is in the interval from 0 to 13,333 and 1/3, you get:
y = 10x = 100,000
y = 80,000 + 8x = 160,000
y = 120,000 + 5x = 170,000
y = 10x is the minimum cost equation in this interval.
at x = 20,000, which is in the interval from 13,333 and 1/3 < x < 24,000, you get:
y = 10x = 200,000
y = 80,000 + 8x = 240,000
y = 120,000 + 5x = 220,000
y = 10x is the minimum cost equation in this interval.
at x = 30,000, which is in the interval from 24,000 < x < 40,000, you get:
y = 10x = 300,000
y = 80,000 + 8x = 320,000
y = 120,000 + 5x = 270,000
y = 120,000 + 5x is the minimum cost equation in this interval.
at x = 50,000, which is in the interval from 40,000 < x < infinity, you get:
y = 10x = 500,000
y = 80,000 + 8x = 480,000
y = 120,000 + 5x = 370,000
y = 120,000 + 5x is the minimum cost eqution in this interval.
what you get is:
y = 10x is the minimum cost equation from 0 < x < 24,000
y = 120,000 + 5x is the minimum cost equation from 24,000 < x < infinity.
the algebraic solution agrees with the graphical solution, as you can determine by looking at the graph and looking at the graphical solution within each interval.
in graph, .....
y = 10x is red
y = 80,000 + 8x is blue
y = 120,000 + 5x is green.