If $y$ varies directly as $x$, and $y$ is 6 when $x$ is 72, what is the value of $y$ when $x$ is 8?
Answer (1)
Establish the direct variation relationship: y = k x .
Determine the constant of proportionality k using the given values x = 72 and y = 6 , which gives k = 12 1 .
Substitute x = 8 into the equation y = 12 1 x .
Calculate the value of y : 3 2 .
Explanation
Understanding Direct Variation 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 = 6 when x = 72 . We can use this information to find the value of k .
Finding the Constant of Proportionality Substitute y = 6 and x = 72 into the equation y = k x to find k : 6 = k ( 72 ) Divide both sides by 72 to solve for k :
k = 72 6 = 12 1 So, k = 12 1 .
Finding y when x = 8 Now that we have the value of k , we can write the equation as y = 12 1 x . We want to find the value of y when x = 8 . Substitute x = 8 into the equation: y = 12 1 ( 8 ) Simplify the expression to find the value of y :
y = 12 8 = 3 2
Examples
Direct variation is a fundamental concept in many real-world scenarios. For instance, the distance you travel at a constant speed varies directly with the time you spend traveling. If you know you travel 100 miles in 2 hours, you can determine your speed (the constant of proportionality) and then calculate how far you'll travel in 3 hours. This principle applies to currency exchange rates, where the amount of one currency you receive varies directly with the amount of the other currency you exchange. Understanding direct variation helps in making predictions and solving problems involving proportional relationships.
