Joint Variation

Review: Direct Variation Direct variation between two variables occurs when one variable is a constant multiple of some power of another one. When one changes, the other changes in the same way: an increase in one causes an increase in the other, and a decrease in one causes a decrease in the other. Example: the area of a circle varies directly with the square of its radius. A = πr2. The constant that relates the two variables is the constant of proportionality, often represented as k. In the previous problem, k = π.

Joint Variation Joint variation occurs when a variable varies directly with two or more variables. An increase in one of the independent variables causes an increase in the dependent variable and a decrease in one of the independent variables causes an decrease in the dependent variable. For instance, if z varies jointly with x and y, we would write z = kxy. k is the constant of proportionality. If we define the relationship in this way, z is the dependent variable because it depends on x and y.

Solving Joint Variation Problems There are three main steps to solving joint variation problems: Translate the given information into an equation. Find the constant of proportionality if it’s not given. Find whatever value you’re looking for.

Example 1 Sometimes joint variable problems will be expressed in terms of variables. For example, say c varies jointly with a and b2. If c =72 when a = 3 and b = 4, what is the constant of proportionality? What is the value of c when b = 2 and a = 5? The first step is to write an equation describing the situation. In this case, we have c = kab2. Now we need to find the constant of proportionality by plugging in known values.

Example 1 (cont.) Plugging in known values, we get 72 = k * 3 * 16, which we can simplify to 48k = 72 When we divide both sides by 48, we get k = 1.5. Our actual relationship, then, is c = 1.5ab2. Now we can find the value of c when a is 5 and b is 2. c = 1.5(5)(2)2 c = 30

Example 2 Sometimes, joint variation problems will describe a real-world situation that you’ll have to translate into mathematical terms. Here’s an example: The volume of a cylinder varies with its height and the square of its radius. If the constant of proportionality is π, what is volume of a cylinder with height 4 inches and radius 3 inches? First, we need to translate this into a mathematical equation. We have all the information we need – constant of proportionality and the relationships between the variables. Our relationship is V = πr2h. To find the volume at h = 4 and r = 3, we plug them into this equation. V = 36π ≈ 113.10