If $y$ varies directly as $x$, and $y$ is 18 when $x$ is 5, which expression can be used to find the value of $y$ when $x$ is 11?
A. $y=\frac{5}{18}(11)$
B. $y=\frac{18}{5}(11)$
C. $y=\frac{(x+5)(5)}{11}$
D. $y=\frac{11}{(18)(5)}$
Answer (2)
Establish the direct variation relationship: y = k x .
Determine the constant of proportionality: k = 5 18 .
Substitute x = 11 into the equation: y = 5 18 ( 11 ) .
The expression to find y when x = 11 is y = 5 18 ( 11 ) .
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 = 18 when x = 5 . We can use this information to find the constant k .
Finding the Constant of Proportionality Substitute y = 18 and x = 5 into the equation y = k x to find k :
18 = k ( 5 ) Divide both sides by 5 to solve for k :
k = 5 18
Finding the Expression for y when x=11 Now that we have found k , we can write the equation relating y and x as: y = 5 18 x We want to find the expression for y when x = 11 . Substitute x = 11 into the equation: y = 5 18 ( 11 )
Final Answer Therefore, the expression to find the value of y when x = 11 is y = 5 18 ( 11 ) .
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 travel 100 miles in 2 hours, your speed is 50 miles per hour. Using direct variation, you can easily calculate how far you'll travel in 3 hours: distance = speed × time = 50 × 3 = 150 miles. This principle applies to various situations, such as calculating the cost of items based on quantity, determining the amount of ingredients needed for recipes, and understanding relationships in physics and engineering.
The expression to find the value of y when x is 11 is y = 5 18 ( 11 ) . The chosen option is B .
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