Question

If \(y\) varies jointly as \(w\) and \(x,\) and \(y\) is 3.85 when \(w\) is 8.36 and \(x\) is \(11.6,\) evaluate the constant of proportionality, and write the complete expression for \(y\) in terms of \(w\) and \(x\).

Short answer

The constant of proportionality is approximately 0.042, and the complete expression for y in terms of w and x is y = 0.042wx.

Step-by-step solution

Understand Joint Variation

Joint variation means that y varies directly as the product of w and x. The equation that describes this relationship is of the form y = kwx, where k is the constant of proportionality.

Plug in Known Values

Substitute the given values into the equation (y = kwx) to find the constant k. We have y = 3.85 when w = 8.36 and x = 11.6. Plug these into the equation: 3.85 = k * 8.36 * 11.6.

Solve for the Constant of Proportionality (k)

Solve for k by dividing both sides of the equation by the product of 8.36 and 11.6: k = 3.85 / (8.36 * 11.6).

Calculate the Constant k

Perform the division to find the value of k: k approx 0.042.

Write the Final Equation for y

Using the value of k, write the final equation that relates y to w and x: y = 0.042wx.

Key concepts

Constant of Proportionality

In algebra, the constant of proportionality is the fixed number that relates two variables that are directly proportional to each other. Think of it as a consistent multiplier that connects an input with an output. For example, if you have a job that pays a constant rate per hour, the amount you earn is directly proportional to the number of hours you work, with the rate per hour being the constant of proportionality.

In the context of joint variation, like in the original exercise, we see multiple inputs interacting, specifically, the variables w and x influencing y. The equation is set up as \(y = kwx\), with k representing this constant. After substituting the known values and performing the calculation, we find that k is approximately 0.042. This means that whenever w and x are multiplied, the product is then multiplied by this constant to yield y.

Direct Variation

Direct variation is a fundamental concept in algebra where one variable changes directly as another variable changes. In other words, if one variable increases, the other increases by a fixed amount, and if one decreases, so does the other. The relationship is described by the equation \(y = kx\), where \(y\) varies directly with \(x\), and \(k\) is the constant of proportionality.

In a joint variation situation, such as that outlined in the exercise, the concept extends to multiple variables. So instead of \(y\) varying in correspondence to just \(x\), it varies directly with the product of \(w\) and \(x\), leading to the equation \(y = kwx\). This stays true to the principles of direct variation in the sense that as the product of \(w\) and \(x\) increases, so does \(y\), in a directly proportional manner, guided by the constant \(k\).

Algebraic Expression

An algebraic expression is a mathematical phrase that includes numbers, variables, and operation symbols, but no equals sign. It can be as simple as \(x + 5\) or as complex as the one seen in our joint variation example. When we describe how \(y\) varies jointly with \(w\) and \(x\), we create the algebraic expression \(y = kwx\).

The process of solving the original problem involved converting a word problem into an algebraic expression using the given relationships and values. After finding the constant of proportionality, we were able to refine our expression to \(y = 0.042wx\), which is a clear, concise algebraic representation of the relationship between \(y\), \(w\), and \(x\). Effective use of algebraic expressions is critical in mathematics as they allow us to succinctly capture and manipulate relationships and change.