Problem 24 - Precalculus Mathematics for Calculus

Question

Express the statement as an equation. Use the given information to find the constant of proportionality. \(t\) is jointly proportional to \(x\) and \(y,\) and inversely proportional to \(r .\) If \(x=2, y=3,\) and \(r=12,\) then \(t=25\).

Step-by-Step Solution

Answer

The constant of proportionality \(k\) is 50. The equation for \(t\) is \(t = 50 \cdot \frac{x \cdot y}{r}\).

Step 1: Understanding Proportionality

The statement "jointly proportional to \(x\) and \(y\), and inversely proportional to \(r\)" forms the basis for expressing \(t\) as an equation. This relationship can be written as \[ t = k \cdot \frac{x \cdot y}{r} \]where \(k\) is the constant of proportionality.

Step 2: Substituting Given Values

We substitute the given values \(x = 2\), \(y = 3\), \(r = 12\), and \(t = 25\) into the equation to find the constant of proportionality \(k\): \[ 25 = k \cdot \frac{2 \cdot 3}{12} \].

Step 3: Solving for the Constant of Proportionality

Simplify the equation by multiplying the values: \[ 25 = k \cdot \frac{6}{12} \].This simplifies further to \[ 25 = k \cdot \frac{1}{2} \].Multiply both sides by 2 to solve for \(k\): \[ 50 = k \].

Step 4: Writing the Final Equation

With the constant of proportionality \(k = 50\), the equation becomes: \[ t = 50 \cdot \frac{x \cdot y}{r} \].

Key Concepts

Joint variation

Joint variation occurs when a variable is directly proportional to the product of two or more other variables. In simpler terms, it means one variable increases when the product of the others increases and vice versa. For example, in our problem, the variable \(t\) is said to be jointly proportional to \(x\) and \(y\). This means that \(t\) changes in accordance with the product of \(x\) and \(y\).
In mathematical terms, joint variation can be expressed as:

• \( t = k \, (x \cdot y) \)

where \(k\) is the constant of proportionality. Here, it's essential to remember that joint variation combines the effects of both variables, multiplying them to see how they together affect \(t\).
So, when you have multiple factors that impact a single outcome, you may be dealing with joint variation. Understanding this can help you decode relationships in various scientific and mathematical contexts.

Inverse variation

Inverse variation happens when one variable decreases as another variable increases. They "move" in opposite directions - one goes up, the other comes down. In our case, \(t\) is inversely proportional to \(r\). This relationship indicates that as \(r\) increases, \(t\) decreases, and vice versa.
The mathematical expression for inverse variation is:

• \( t = \frac{k}{r} \)

This form demonstrates that \(t\) and \(r\) are inversely related, meaning they will affect each other in reverse ways. Typically, multiplying them together gives a constant. Here, inverse variation with \(r\) means as \(r\) grows larger, the value of \(t\) shrinks for the same product of \(k\).
Seeing this frequently helps identify where an increase in one factor leads to a proportional decrease in another, which can be insightful for understanding different scenarios in physics or economics.

Constant of proportionality

The constant of proportionality \(k\) is a crucial factor in variation equations. It acts as a bridging number that connects the variables in their proportional relationships. In our problem, once the values for \(t, x, y,\) and \(r\) are known, \(k\) can be calculated to help complete the equation.
To find \(k\), you'd follow these steps:

• Substitute the given values into the joint and inverse variation formula: \[ t = k \cdot \frac{x \cdot y}{r} \]
• Plug in the known values here: \(25 = k \cdot \frac{2 \cdot 3}{12}\)
• Solve for \(k\) by simplifying and multiplying: \(k = 50\)

With \(k = 50\), it finalizes the relationship between the variables using this specific constant. The constant of proportionality shows the exact impact one set of factors has on their product, helping to precisely measure and predict outcomes in many practical situations.