Problem 9 - College Algebra

Question

Use the four-step procedure for solving variation problems given on page 424 to solve. \(y\) varies jointly as \(a\) and \(b\) and inversely as the square root of \(c, y=12\) when \(a=3, b=2,\) and \(c=25 .\) Find \(y\) when \(a=5, b=3,\) and \(c=9\)

Step-by-Step Solution

Answer

The new value of \(y\) when \(a = 5\), \(b = 3\), and \(c = 9\) is \(200\).

Step 1: Identify the Equation for Variation

As said in the instruction, 'y' varies jointly as 'a' and 'b', and 'y' varies inversely as the square root of 'c'. Therefore, the general equation for this variation is \(y = k \cdot a \cdot b / \sqrt{c}\), where \(k\) is the constant of variation.

Step 2: Find the Constant of Variation

Insert the initial conditions into the equation from Step 1, \(a = 3\), \(b = 2\), and \(c = 25\), and solve for \(k\). \(12 = k \cdot 3 \cdot 2 / \sqrt{25}\). Solving for \(k\), we obtain \(k= 40\)

Step 3: Use the constant of Variation to find y

With the constant of variation \(k = 40\) found in Step 2, we can insert the new values for a, b, and c into the equation from step 1. Therefore, \(y = 40 \cdot 5 \cdot 3 / \sqrt{9}\). Solving this expression, \(y = 200\)

Key Concepts

Joint Variation

Joint variation is a concept where a variable is directly proportional to the product of two or more other variables. In simpler terms, if one variable increases, the other related variables increase proportionally. This relationship can be seen in the formula:

• For joint variation: \( y = k \cdot a \cdot b \)
• Here, \( y \) varies jointly with the product of \( a \) and \( b \).

The constant \( k \) represents how strongly \( y \) is related to the product of \( a \) and \( b \). In our exercise, \( y \) depends on \( a \) and \( b \) in this manner. This means, as \( a \) or \( b \) increases, \( y \) would also increase, provided \( c \) remains constant.
Joint variation can be found in real-world scenarios like calculating work done, where work is proportional to the product of force and distance.

Inverse Variation

Inverse variation occurs when an increase in one variable leads to a decrease in another variable. In these cases, the variables are inversely related, meaning as one becomes larger, the other becomes smaller.

• The general formula for inverse variation is \( y = k / x \).
• As seen in the exercise, \( y \) varies inversely as the square root of \( c \).

This means as \( c \) gets larger the square root of \( c \) increases, and \( y \) becomes smaller, assuming \( a \) and \( b \) remain constant. In the given problem, the inverse relationship keeps \( y \) from getting too large as \( c \) influences the outcome.
Inverse variations are present in many fields, like physics, for example with the concept of gravitational force getting weaker with increased distance.

Constant of Variation

The constant of variation, denoted as \( k \), is a vital component in both joint and inverse variation equations. It acts as a proportionality factor connecting the variables in the relationship.

• In the joint variation case, \( y = k \cdot a \cdot b \).
• For inverse variation, we have \( y = k / \sqrt{c} \).

The constant \( k \) ensures the equation balances out, making sure the values harmonize perfectly. In the given exercise, solving for \( k \) was done by using known variables \( a \), \( b \), and \( c \) to assess the initial condition \( y = 12 \). Once identified, \( k = 40 \) was used to solve for new values.
Understanding the constant of variation helps in identifying how strong the dependence of variables is upon each other under given conditions.

Solving Equations

Solving equations involving variation principles requires a methodical approach. Here’s a simplified approach:

• Step 1: Identify the general form of the equation reflecting the variation type - joint, inverse, or a combination.
• Step 2: Insert known values into the equation to solve for the constant \( k \).
• Step 3: Substitute new values along with the calculated \( k \) into the equation to find the unknown.

This approach helps in breaking down complex relationships into simpler, solvable steps. In the exercise provided, this procedure assisted in solving for \( y \) by first determining the constant and then using it to find \( y \) with new inputs.
Solving such variation equations reinforces basic algebra skills and logical reasoning, fostering a deeper understanding of relationships among variables.