Question

write a general formula to describe each variation. \(z\) varies directly with the sum of the squares of \(x\) and \(y ; z=26\) when \(x=5\) and \(y=12\)

Short answer

The general formula is \( z = \frac{2}{13}(x^2 + y^2) \).

Step-by-step solution

Understand the Variation

Direct variation means that as one variable changes, the other variable changes in a proportional manner. For the given problem, this translates to the equation: \[ z = k(x^2 + y^2) \]where \(k\) is the constant of proportionality.

Substitute Given Values

Substitute the known values into the equation. Given \(z = 26\), \(x = 5\), and \(y = 12\), we have: \[ 26 = k(5^2 + 12^2) \]which simplifies to: \[ 26 = k(25 + 144) \]\[ 26 = k(169) \]

Solve for the Constant \(k\)

Isolate \(k\) by dividing both sides by 169: \[ k = \frac{26}{169} \]This simplifies to: \[ k = \frac{2}{13} \]

Write the General Formula

Substitute the constant \(k\) back into the original variation equation: \[ z = \frac{2}{13}(x^2 + y^2) \]

Key concepts

constant of proportionality

In direct variation, two variables change in such a way that one variable is always a constant multiple of the other.
This constant multiple is known as the 'constant of proportionality'.
Let's look at an example where variable z varies directly with the sum of the squares of variables x and y.
This relationship can be written as: \[ z = k(x^2 + y^2) \] Here, 'k' is the constant of proportionality.
To find the exact value of 'k', we need some known values of x, y, and z.
In the exercise, we were given z=26 when x=5 and y=12.
We substitute these values into the equation to solve for 'k': \[ 26 = k(5^2 + 12^2) \] Simplifying the sum of squares: \[ 26 = k(25 + 144) \] \[ 26 = k(169) \] We can now isolate 'k' by dividing both sides by 169: \[ k = \frac{26}{169} = \frac{2}{13} \] Thus, the constant of proportionality for this variation problem is \( k = \frac{2}{13}) \).

squares of variables

In this problem, the relationship involves the squares of the variables x and y.
Squaring a number means multiplying that number by itself.
For example, the square of 5 is: \[ 5^2 = 5 \times 5 = 25 \] Similarly, for y = 12: \[ 12^2 = 12 \times 12 = 144 \] When we add these squares, we get: \[ 5^2 + 12^2 = 25 + 144 = 169 \] This sum of squares is then used in our variation equation.
Notice that the effect of squaring the variables and then adding them before applying the constant 'k' takes into account the combined influence of x and y on z.
It is significant because, unlike direct proportionality with individual variables, this relationship magnifies the impact when either x or y increases.

general formula derivation

Deriving the general formula in a variation problem involves identifying the relationship between the variables and expressing it mathematically.
Here, since z varies directly with the sum of the squares of x and y, we start with the equation: \[ z = k(x^2 + y^2) \] Given the values of x, y, and z, we substitute these into the equation to find the constant 'k'.

• Given: z = 26, x = 5, y = 12
• Substitute: \[ 26 = k(5^2 + 12^2) \]
• Simplify: \[ 26 = k(169) \]
• Solve: \[ k = \frac{26}{169} = \frac{2}{13} \]

Now that we have 'k', we return to the original formula and place the value of 'k' back into it: \[ z = \frac{2}{13}(x^2 + y^2) \] This general formula can now be used to find 'z' for any values of x and y, as long as the relationship of direct variation described in the problem is maintained.