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 (2)
Find the constant of proportionality k using the given values x = 72 and y = 6 in the direct variation equation y = k x , which gives k = 72 6 = 12 1 .
Substitute x = 8 and k = 12 1 into the equation y = k x to find the value of y .
Calculate y = 12 1 × 8 = 12 8 = 3 2 .
The value of y when x = 8 is 3 2 .
Explanation
Finding the constant of proportionality 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 .
Calculating k Substitute y = 6 and x = 72 into the equation y = k x to get 6 = k ( 72 ) . To solve for k , we divide both sides of the equation by 72: k = 72 6 = 12 1 So, the constant of proportionality is 12 1 .
Finding y when x=8 Now we want to find the value of y when x = 8 . We use the equation y = k x with k = 12 1 and x = 8 . Substituting these values into the equation, we get y = 12 1 \t × 8 = 12 8 = 3 2 Thus, when x = 8 , y = 3 2 .
Final Answer Therefore, the value of y when x is 8 is 3 2 .
Examples
Direct variation is a concept that appears 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 constant speed of 60 miles per hour, the distance d you travel is given by d = 60 t , where t is the time in hours. This means that if you double the time you spend traveling, you double the distance you travel. Similarly, the amount you earn at an 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 making predictions and understanding relationships between quantities in various practical situations.
To find the value of y when x is 8, we first calculate the constant of proportionality, k, using the initial values. This gives us k = 1/12. Substituting k into the direct variation equation for x = 8 results in y = 2/3.
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