Direct variation can be used to model real-world situations.
Many real-world situations can be modeled by direct variation. In geometry, the perimeter of a square is proportional to the side length of the square by the formula . The formula describes how distance is proportional to time by the constant of proportionality , which represents a constant rate of travel. When shopping for several same-price items, the total cost is proportional to the number of items by the price of each item, or .Sales tax is another example of direct variation. The amount of sales tax on an item is proportional to the price. If the sales tax rate is , then the amount of sales tax on an item with price is . The total cost of the item with tax is:
Another application of direct variation is converting units, such as distance, volume, or currency. The constant of variation is the conversion factor.
Applying Direct Variation to Sales Tax
Graph and then determine the exact total cost of an item with a price of $5 after a sales tax of 10%.
The total cost of an item with price after a sales tax rate of is represented by the sales tax equation:
For a 10% sales tax, the value of is 0.10, which can also be written as 0.1. Substitute 0.1 for .
Determine two points on the line of the equation.The graph of a direct variation passes through the point and , where is the constant of proportionality. In this situation, is 1.1.
So, the line of the equation goes through the points and .
Plot the points and determine the slope, or , of the equation. The graph of the equation goes through the origin with a slope of 1.1. From the graph, when the price of the item is $5, the total cost after tax is between $5 and $6. Substitute 5 for in the equation to identify the exact total cost.
The exact total cost is $5.50.
Applying Direct Variation to Currency
Currency in U.S. dollars varies directly with the value in other currencies. On a given day, 8 euros are exchanged for 9.25 U.S. dollars. How many euros would be exchanged for 50 U.S. dollars?
Write the direct variation equation. Let represent the number of euros and represent the number of U.S. dollars.
Substitute the given values of and into the equation, and solve for .
This means that the exchange rate is about 0.865 euro per U.S. dollar.
Substitute the value of into the direct variation equation.
The number of U.S. dollars is given, which is represented by . Substitute 50 for the value of into the equation.
At the given exchange rate, 43.25 euros would be exchanged for 50 U.S. dollars.