Basic Linear Models

Example 1

If a person pays $45.50 in sales tax on items with value of $910, write an equation for the amount of tax paid in purchasing items worth x dollars.

Solution .

Since the tax varies directly with the dollar amount purchased, y = mx. Letting y = 45.5 and x = 910, we get 45.5 = 910m. Solving this, we have m = 0.05. Thus, y = 0.05x

Example 2

If the fixed cost to produce an item is $15000 and the total cost to produce 100 items is $17500, then find the total cost function. Assume that total cost is a linear function of the number of items.

Solution .

Since the fixed cost is the cost to produce zero items, the data given corresponds to two points on the linear function. The points are (0,15000) and (100,17500). Using the definition of slope, . This give us . Since (0,15000) is the y-intercept, we have the equation which give us y (cost) as a function of x (number of items).

Note: In this example the slope of 25 is actually $25 per item. This is sometimes referred to as the variable cost.

Example 3

A company purchases a piece of equipment for $800. In 5 years the equipment will have no value. Write a linear equation giving the value of the equipment during the 5 year period.

Solution .

Two points given on the line are (0, 800) and (5, 0). The slope is then or -160 which means that the equipment loses $160 of value per year. Since (0, 800) is the y-intercept the equation is

The following table shows the value of the equipment at the end of each year of its life.

 x            v(x)    

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

0            800.       

1.000000000  640.0000000

2.000000000  480.0000000

3.000000000  320.0000000

4.000000000  160.0000000

5.000000000  0          

This is known as stright line depreciation.

Example 4

A warehouse has 1300 17 inch super VGA monitors. The warehouse ships 182 of these monitors in a seven day period. Assuming that this rate continues (and the wharehouse receives no monitors), find the equation giving the number of monitors in the warehouse. Make a graph over the interval for which the number of monitors is nonnegative.
Solution .

Since the number of monitors in the warehouse is decreasing, the slope of the line is or -26. (i.e. the number of monitors in the wharehouse is decreasing by 26 per day.) Because there are 1300 monitors initially, b = 1300. The number of monitors (y) in the wharehouse is given by where x is in days.

To locate the interval over which the number of monitors is nonnegative, we must find when y = 0. (i.e. find the x-intercept) Solving -26x + 1300 = 0 gives us x = 50.