$x$ varies directly as the product of $u$ and $v$ and inversely as their sum. If $x=2$ when $u=3$ and $v=1$, what is the value of $x$ if $u=3$ and $v=3$?
Answer (1)
Establish the relationship: x = k u + v uv .
Solve for k using given values: 5 = k 3 + 1 3 ( 1 ) , which gives k = 3 20 .
Substitute k , u , and v to find x : x = 3 20 3 + 3 3 ( 3 ) .
Simplify to find the value of x : 10 .
Explanation
Formulate the equation We are given that x varies directly as the product of u and v and inversely as their sum. This can be written as: x = k u + v uv where k is the constant of proportionality.
Find the constant of proportionality We are given that x = 5 when u = 3 and v = 1 . We can use this information to find the constant of proportionality k . Substituting these values into the equation, we get: 5 = k 3 + 1 3 ( 1 ) 5 = k 4 3
Calculate k Solving for k , we multiply both sides of the equation by 3 4 :
k = 5 ⋅ 3 4 = 3 20
Calculate x Now we want to find the value of x when u = 3 and v = 3 . We substitute these values and the value of k into the equation: x = 3 20 3 + 3 3 ( 3 ) x = 3 20 6 9 x = 3 20 2 3 x = 10
Examples
Understanding direct and inverse variations is crucial in many real-world scenarios. For instance, in physics, the force of gravitational attraction between two objects varies directly with the product of their masses and inversely with the square of the distance between them. Similarly, in economics, the demand for a product might vary directly with advertising expenditure and inversely with the price of the product. By mastering these concepts, you can model and analyze various phenomena in science, economics, and engineering.
