Question

write a general formula to describe each variation. \(z\) varies directly with the sum of the cube of \(x\) and the square of \(y ; \quad z=1\) when \(x=2\) and \(y=3\)

Short answer

The general formula is \(z = \frac{1}{17}(x^3 + y^2)\).

Step-by-step solution

Identify the relationship

Since the problem states that \(z\) varies directly with the sum of the cube of \(x\) and the square of \(y\), we can express this as: \(z = k(x^3 + y^2)\). Here, \(k\) is the constant of proportionality.

Find the constant of proportionality

Given that \(z=1\) when \(x=2\) and \(y=3\), substitute these values into the equation \(z = k(x^3 + y^2)\):\(1 = k(2^3 + 3^2)\) which simplifies to \(1 = k(8 + 9)\) or \(1 = k \times 17\). Solving for \(k\) gives: \(k = \frac{1}{17}\).

Write the general formula

Now that we know the constant of proportionality \(k\), we can write the general formula describing the variation: \(z = \frac{1}{17}(x^3 + y^2)\).

Key concepts

Constant of Proportionality

In direct variation problems, the constant of proportionality, represented by the letter 'k', plays a key role. This constant defines how one quantity changes in relation to another. For example, if one variable doubles, the other variable also doubles if they are directly proportional. To find 'k', you typically need a pair of corresponding values for the variables involved. These are used in a given formula to solve for this constant. For the exercise provided, knowing that both the cube of x and square of y contribute to the value of z helps us understand their proportional relationship.

Direct Variation Formula

The direct variation formula is an equation that shows the relationship between variables that vary directly. It is generally expressed as:

• \text{\{z = k(x^3 + y^2)\}}

In this context, 'z' varies directly with the sum of the cube of 'x' and the square of 'y', and 'k' is the constant of proportionality. This formula tells us that as the sum of \left\(x^3 + y^2\right\) increases, the value of 'z' will changes accordingly. It's essential to identify proper formulas based on the relationships described in the problem to solve for unknowns efficiently.

Solving for Constants

To find the constant of proportionality 'k', you need specific values for the variables. In the given problem, we start with values given for 'x' and 'y', which are 2 and 3, respectively. Substituting these into our direct variation formula helps us solve for 'k'.

• First, calculate the terms inside the parenthesis: \left\(x^3 + y^2\right\)
• Replace the values: 1 = k (2^3 + 3^2)
• Simplify the equation to find 'k': 1 = k (8 + 9) = k (17)

So in this case, solving for 'k' involves simple arithmetic to isolate 'k'. The value of 'k' becomes 1 divided by 17, or \frac{1}{17}\. With 'k' found, we can write the general formula: z = \frac{1}{17}(x^3 + y^2)\. This constant allows us to relate 'z' to new values of 'x' and 'y' in a consistent way.