$x$ varies directly as the product of $u$ and 4 and inversely as their sum. If $x = 3$ when $u = 3$ and $v = 1$, what is the value of $x$ if $u = 3$ and $v = 3$?
Answer (1)
Write the equation for x in terms of u and v : x = k u + v uv .
Substitute x = 3 , u = 3 , and v = 1 into the equation to find the value of k : 3 = k 3 + 1 3 ( 1 ) .
Solve for k : k = 4 .
Substitute k = 4 , u = 3 , and v = 3 into the equation to find the value of x : x = 4 3 + 3 3 ( 3 ) = 6 . The final answer is 6 .
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.
Substitute Given Values We are given that x = 3 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: 3 = k 3 + 1 3 ( 1 ) 3 = k 4 3
Solve for k To solve for k , we multiply both sides of the equation by 3 4 : k = 3 × 3 4 = 4 So, k = 4 .
Find x with New Values Now we want to find the value of x when u = 3 and v = 3 . We substitute these values and k = 4 into the equation: x = 4 3 + 3 3 ( 3 ) x = 4 6 9 x = 4 2 3 x = 6
Final Answer Therefore, the value of x when u = 3 and v = 3 is 6.
Examples
Understanding direct and inverse variations is crucial in many real-world scenarios. For instance, in physics, the gravitational force 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 a wide range of phenomena.
