Problem 25 - Algebra 2

Question

Write the function that models each variation. Find \(z\) when \(x=4\) and \(y=9\) \(z\) varies jointly with \(x\) and \(y .\) When \(x=2\) and \(y=3, z=60\)

Step-by-Step Solution

Answer

The value of z when x=4 and y=9 is 360.

Step 1: Calculate the Joint Variation Constant k

Start by setting up the formula with the original conditions, with z=60, x=2, and y=3. Plug these values into the formula \(z=kxy\). This gives:60 = k \cdot 2 \cdot 3. Now, solve for k, which is 60 / (2*3) = 10

Step 2: Find z using x=4 and y=9

Next, replace k, x and y in the original formula with the new discovered values. This gives \(z = 10*4*9\). Solving it will yield to the answer.

Key Concepts

Variation Constant

When discussing joint variation, a crucial component is the variation constant, often symbolized as \(k\). This constant is a fixed number that helps relate the variables in the joint variation equation. In the problem we're solving, the equation is represented as \(z = kxy\), meaning \(z\) varies jointly with \(x\) and \(y\).
To find \(k\), you need known values of \(z\), \(x\), and \(y\). In our example, the values are \(z = 60\), \(x = 2\), and \(y = 3\).
Substitute these values into the equation to solve for \(k\). Doing this, we get:

• \(60 = k \cdot 2 \cdot 3\)
• Solve for \(k\) by dividing both sides by \(6\) (since \(2 \cdot 3 = 6\))
• \(k = \frac{60}{6} = 10\)

Now, \(k = 10\) is your variation constant, and it helps you to predict \(z\) when \(x\) and \(y\) change.

Direct Variation

Direct variation is a key part of understanding joint variation. In direct variation, a variable changes proportionally with another variable through a constant multiplier. This means if one variable increases, the other increases proportionally, assuming a positive constant.
Mathematically, you’d express direct variation with a formula like \(z = kx\), where \(k\) is a positive constant. In joint variation, this concept extends to multiple variables, such as \(z = kxy\).
So, in our problem, when \(z = kxy\), we see \(z\) directly varies with both \(x\) and \(y\) at the same time. If you know \(k\), and any two variables out of \(z\), \(x\), or \(y\), you can always find the third. The relationship keeps direct variation as its core but expands it to include more factors. Understanding this helps when manipulating variables in real-life scenarios where several factors may influence an outcome jointly.

Jointly with Multiple Variables

Joint variation means that a variable is related to two or more other variables at the same time, and they together are influencing a particular situation. When \(z\) is described as jointly varying with \(x\) and \(y\), it illustrates that changes in \(z\) arise from simultaneous changes in \(x\) and \(y\). Using joint variation can be essential in real-life applications where multiple elements affect the outcome.
Here is how the equation \(z = kxy\) works:

• The variable \(z\) increases when either \(x\) or \(y\) increases, thanks to the constant \(k\).
• If both \(x\) and \(y\) double, \(z\) quadruples, which shows the power of compounding effects.
• When \(z = 10 \cdot 4 \cdot 9 = 360\), it demonstrates this influence -- with \(k = 10\), the calculations capture how strategy changes in \(x\) or \(y\) can create significant shifts in \(z\).

This concept is invaluable anytime you need to predict outcomes based on the simultaneous change of several factors, useful in fields like economics, physics, and biology.