Thread: linear function...??

I've tried to simplify the following example as much as possible for ease of reading...



A manufacturer finds that for a new model, at £240 a unit, 20 units per week are sold. However, during promotion, the model was priced at £160 per unit, 45 units per week were sold. Let S be number of units sold per week, and X (in GBP) be the price of a unit. All production is sold.

How do I show that, for value of the price X, the number sold S is given
by the linear function:



S = 95 - (5/16)X


Any help much appreciated, Thank you.

Hi

It is assumed that S is a linear function of X
S = aX + b

From the data
20 = 240 a + b
45 = 160 a + b

By difference
-25 = 80 a => a = -25/80 = -5/16
b = 45 - 160 a = 45 + 50 = 95

Originally Posted by bobchiba

A manufacturer finds that for a new model, at £240 a unit, 20 units per week are sold. However, during promotion, the model was priced at £160 per unit, 45 units per week were sold. Let S be number of units sold per week, and X (in GBP) be the price of a unit. All production is sold.

How do I show that, for value of the price X, the number sold S is given
by the linear function...

You've been given two data points: (X, S) = (240, 20) and (X, S) = (160, 45). Plug these two points into the formula for the slope of the line through these points.

Pick one of the points (it doesn't matter which), and plug it and the slope into one of the forms of the linear equation, such as the point-slope form.

Solve for "S=" to get your answer.

Ok, thanks guys, looking at the continuation of the example, this is the next stage...



There are production costs of £80 per unit and fixed costs of £1000

per week. Explain why the total cost C per week can be expressed as

C= 1000 + 80S



and hence show that in terms of the price C = 8600 - 25X


I am right in saying this is quadratic, and not linear?

Originally Posted by bobchiba

looking at the continuation of the example, this is the next stage...

There are production costs of £80 per unit and fixed costs of £1000 per week. Explain why the total cost C per week can be expressed as C= 1000 + 80S and hence show that in terms of the price C = 8600 - 25X

I am right in saying this is quadratic, and not linear?[/FONT]

You have the number sold S as a function of the unit price X, with the function being of the form S = 95 - (5/16)X. Now you are asked about the costs.

The "fixed" costs are the costs of "just showing up"; they are unrelated to how many units you have to produce for sale. For instance, the rent on your store-front will be the same, regardless of whether you have any customers in a given month.

The "production" costs are the costs for producing each unit. If you produce 1 unit, your production cost will be 1(80) = 80 pounds. If you produce 2 units, your production cost will be 2(80) = 160 pounds. If you produce S units, what then will be your production costs?

The total cost C is the sum of the variable ("production") costs and the fixed costs. Use this fact to find your "cost" relation.

Once you have shown that C = 1000 + 80S, use the first equation to replace S. Since "S" is defined to be the same as "95 - (5/16)X", plug "95 - (5/16)X" in for "S" in "C = 1000 + 80S". Simplify to get the desired result.

Note: The expression for C will indeed be what the book said, and not a quadratic.