If $y$ varies directly as $x$, and $y$ is 20 when $x$ is 4, what is the constant of variation for this relation?
Answer (1)
Recognize that direct variation implies a relationship of the form y = k x .
Substitute the given values x = 4 and y = 20 into the equation.
Solve for the constant of variation k by dividing both sides by 4: k = 4 20 = 5 .
The constant of variation is 5 .
Explanation
Understanding the Problem We are given that y varies directly as x . This means that there is a constant k such that y = k x . We are also given that y = 20 when x = 4 . We need to find the value of k , which is the constant of variation.
Substituting the Values We can substitute the given values of x and y into the equation y = k x to find the constant of variation k . Substituting x = 4 and y = 20 , we get 20 = k × 4
Solving for k To solve for k , we divide both sides of the equation by 4: 4 20 = 4 k × 4 5 = k
Finding the Constant of Variation Therefore, the constant of variation is 5.
Examples
Direct variation is a fundamental concept in many real-world scenarios. For example, the distance you travel at a constant speed varies directly with the time you spend traveling. If you travel at a speed of 60 miles per hour, the distance d you travel is given by d = 60 t , where t is the time in hours. Similarly, the amount you earn at a fixed hourly rate varies directly with the number of hours you work. If you earn 15 p er h o u r , yo u re a r nin g s e a re g i v e nb y e = 15h , w h ere h$ is the number of hours you work. Understanding direct variation helps in predicting outcomes based on constant relationships.
