PhD. in Mathematics
Norm was 4th at the 2004 USA Weightlifting Nationals! He still trains and competes occasionally, despite his busy schedule.
The average cost is the cost per item of producing a certain number of items, or c(x)/x. So, to find average cost for a certain, given a cost function, simply plug in the number of items into the cost function, and divide by that number. The result is the cost per item produced.
Let's take the idea of cost a little further. Average cost is the cost per item of producing x items. So average cost would be the total cost function, c(x), divided by x, the number of items produced. Let's do this in a problem.
The problem says your magic broom company computes total cost using this linear function; c(x) equals 300,000 Galleons plus 55x. Find the average cost if a 1000 brooms are produced.
So you would calculate average cost as c(1000) divided by 1000, just from this formula. So I calculate c(1000), that's 300,000, plus 55 times a 1000, that's 55,000 all that over 1,000. That's 355,000 over a 1,000, 355 Galleons per broom. That's the average cost of producing 355 galleons per broom.
Now b; your best goblin alerts you that you have been using the wrong cost funcion. It should be c(x) equals 300,000 plus 60x minus 0.03x² plus 0.000009x³. A cubic function. It looks a little bit more like this. This blue one here. The one you've been using is this linear one. So this reflects how you buy more materials, the cost goes down. You buy too much, resources become scarce, the cost goes up. Find the average cost if 1,000 brooms are produced.
So again we're calculating c(1,000) divided by 1,000. Only this time, we're using this massive function to get the cost. So I need 300,000 plus 60 times 1,000 that's 60,000, minus 0.03 times a 1,000². That's a million times 0.03. That's minus 30,000 plus 0.000009 times 1,000³ is a billion. So this is going to be 9 times 10 to the -6, times a billion, that's 9,000. So all that over 1,000.
So I have 360,000 minus 30,000, 330,000 plus 9,000; 339,000. That's the total cost of producing 1000 brooms. I have to divide that by the 1000. That gives me 339 galleons per broom. So average cost is a way of measuring cost per broom produced, or per item produced.