When [tex]$y$[/tex] is 4, [tex]$p$[/tex] is 0.5, and [tex]$m$[/tex] is 2, [tex]$x$[/tex] is 2. If [tex]$x$[/tex] varies directly with the product of [tex]$p$[/tex] and [tex]$m$[/tex] and inversely with [tex]$y$[/tex], which equation models the situation?
[tex]$\frac{x y}{p m}=8$[/tex]
[tex]$\frac{x p m}{y}=0.5$[/tex]
[tex]$\frac{x}{p m y}=0.5$[/tex]
Answer (1)
The problem states that x varies directly with p and m and inversely with y , so we write the equation x = k y p m .
We substitute the given values x = 2 , p = 0.5 , m = 2 , and y = 4 into the equation to find k : 2 = k 4 ( 0.5 ) ( 2 ) .
We solve for k : k = 8 .
We substitute k = 8 back into the equation and rearrange to find the equation that models the situation: p m x y = 8 .
The final answer is p m x y = 8
Explanation
Formulate the Basic Equation We are given that x varies directly with the product of p and m and inversely with y . This means that x is proportional to y p m . We can write this relationship as: x = k y p m where k is the constant of proportionality.
Substitute Given Values We are given that when y = 4 , p = 0.5 , m = 2 , and x = 2 . We can substitute these values into the equation to find the constant of proportionality k : 2 = k 4 ( 0.5 ) ( 2 ) 2 = k 4 1
Solve for k To solve for k , we multiply both sides of the equation by 4: 2 × 4 = k 4 1 × 4 8 = k So, k = 8 .
Write the Final Equation Now that we have found the value of k , we can write the equation that models the situation: x = 8 y p m To match one of the given options, we can rearrange the equation. Multiply both sides by y : x y = 8 p m Divide both sides by p m : p m x y = 8
Examples
Understanding direct and inverse variations is crucial in many real-world scenarios. For instance, in physics, the speed of an object might vary directly with the force applied and inversely with its mass. Similarly, in economics, the demand for a product might vary directly with consumer income and inversely with the product's price. These relationships help us predict and analyze how changes in one variable affect others, allowing for better decision-making and planning.
