Problem 20 - Beginning and Intermediate Algebra

Question

Suppose \(A\) varies directly as \(D .\) If \(A=12\) when \(D=3\) a) find the constant of variation. b) write the specific variation equation relating \(A\) and \(D\) c) find \(A\) when \(D=11\)

Step-by-Step Solution

Answer

a) The constant of variation is \(k = 4\). b) The specific variation equation relating \(A\) and \(D\) is \(A = 4D\). c) When \(D = 11\), \(A = 44\).

Step 1: Find the constant of variation

Since \(A\) varies directly as \(D\), we can write the equation as \(A = kD\). We are given a specific case where \(A = 12\) and \(D = 3\). Let's substitute these values into the equation and solve for the constant of variation \(k\). \(12 = k \cdot 3\) Now, divide both sides by 3 to solve for \(k\): \(k = 4\) So, the constant of variation is 4.

Step 2: Write the specific variation equation

Now that we have the constant of variation, \(k=4\), we can write the specific variation equation relating \(A\) and \(D\). Simply substitute the value of \(k\) into the general equation: \(A = 4D\) This is the specific variation equation relating \(A\) and \(D\).

Step 3: Find A when D = 11

We are given that \(D = 11\), and we need to find the value of \(A\). Using the specific variation equation we found in step 2: \(A = 4 \cdot 11\) Multiply 4 by 11: \(A = 44\) So, when \(D = 11\), the value of \(A\) is 44.

Key Concepts

Constant of Variation

In direct variation, the constant of variation is a special number that connects two variables. Think of it as a magical multiplier that shows how one variable increases or decreases in relation to another. In our exercise, we see that the relationship between variables \(A\) and \(D\) is given by the equation \(A = kD\). Here, \(k\) is the constant of variation.

To find this constant, we use known values of \(A\) and \(D\). For instance, when \(A = 12\) and \(D = 3\), substituting these into our equation gives us \(12 = k \cdot 3\). By solving for \(k\), we divide both sides by 3, resulting in \(k = 4\).

• The constant of variation remains the same for any consistent relationship.
• It helps us predict one variable when we know the other.

Understanding this principle is key to solving problems involving direct variation.

Variation Equation

The variation equation is a simple linear equation that relates two variables using the constant of variation. It's like a simple rule you can follow to map out their relationship. In the previous step, we discovered that the constant of variation \(k\) is 4.

Now, we can rewrite the general direct variation formula \(A = kD\) with our specific constant. This gives us a specific equation: \(A = 4D\). This equation tells us how \(A\) changes whenever \(D\) changes by a specific factor.

• This equation captures the consistent relationship between the variables.
• It enables us to calculate different values for \(A\) given various values of \(D\).

With every value of \(D\), the equation will give a corresponding value for \(A\), using \(k = 4\) as the direct link.

Solving for Variables

Solving variable problems in a direct variation equation is about using your variation equation to find unknowns. Once you have determined or have been given the constant of variation and specific equation, the rest becomes straightforward. Let's see how we apply this.

Given the equation \(A = 4D\), to find \(A\) when \(D = 11\), we substitute \(D\) into our equation: \(A = 4 \times 11\). By performing the multiplication, we see that \(A = 44\).

• To solve for \(A\), plug in the known value of \(D\) into the equation.
• For each problem, ensure you solve using the variable you're asked to find.

This step-by-step substitution process makes solving these problems easier and more intuitive, allowing you to find any variable whenever one is missing.