Question

write a general formula to describe each variation. \(M\) varies directly with the square of \(d\) and inversely with the square root of \(x ; M=24\) when \(x=9\) and \(d=4\)

Short answer

The general formula is \( M = 4.5 \cdot \frac{d^2}{\sqrt{x}} \).

Step-by-step solution

- Identify the relationship

The problem states that variable M varies directly with the square of d and inversely with the square root of x. This can be formulated as: \[ M = k \cdot \frac{d^2}{\sqrt{x}} \] where k is the constant of proportionality.

- Substitute the given values

Substitute the given values of M, d, and x into the equation to find the constant k: \[ 24 = k \cdot \frac{4^2}{\sqrt{9}} \]

- Simplify the equation

Simplify the values within the equation: \[ 24 = k \cdot \frac{16}{3} \]

- Solve for k

Solve for k: \[ k = 24 \cdot \frac{3}{16} \] \[ k = 4.5 \]

- Write the general formula

Now that k is known, substitute it back into the original formula: \[ M = 4.5 \cdot \frac{d^2}{\sqrt{x}} \]

Key concepts

proportionality constant

The proportionality constant, often represented as 'k', links variables in direct or inverse variation equations. In direct variation, as one variable increases, the other does as well. Inverse variation means one variable increases while the other decreases. To find 'k', substitute given values into the equation. Here, in the problem we were solving, the constant 'k' indicated how much M would vary with respect to the square of 'd' and the square root of 'x'. By substituting the known values (M=24, d=4, x=9), we solved for 'k'. Finding 'k' gives us the ability to create a general formula applicable to other values for these variables.

solving equations

Solving equations involves simplifying and manipulating them to find the value of an unknown variable. The equation represented the relationship between M, d, and x: Μετατροπή πραγματικού στις σωστές μορφές . For our problem, we simplify and solve step-by-step to isolate 'k'. Starting with the direct and inverse variation formula, we substitute the given values and simplify each part until 'k' stands alone. This step-by-step process ensures clarity and accuracy. Solve equations one part at a time, checking each step can prevent mistakes. First, we substitute, then simplify arithmetic expressions: Overall, it involves:

• Identifying the given variables and constants in the problem
• Substituting these values into the general formula
• Simplifying the equation step-by-step
• Solving for the unknown variable

variation problems

Variation problems describe how one variable changes in response to another. They come as direct, inverse, or combined variations. Direct variation occurs when two variables change in the same way. With inverse variation, one increases while the other decreases. Combined variation involves both direct and inverse relationships. Problems can describe more complex relationships as in our problem's requirement:

• Direct with the square of one variable
• Inverse with the square root of another

Variation problems help understand relationships between quantities effectively. By setting up an equation based on these relationships, we can predict how changes in one quantity affect another. This is useful in science, economics, and everyday scenarios where understanding proportionality leads to better decisions and insights:Typically entails:

• Identifying the type of variation (direct/inverse/combined)
• Writing the appropriate formula
• Solving for constants or other variables using given values

It explains how variables intertwine, providing clear, predictable relationships between them.