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. $y=\frac{(18)(5)}{11}$
D. $y=\frac{11}{(18)(5)}$
Answer (2)
Express the direct variation as y = k x .
Find the constant of proportionality k using the given values x = 5 and y = 18 , which gives k = 5 18 .
Substitute x = 11 into the equation y = 5 18 x .
The expression to find y when x = 11 is y = 5 18 ( 11 ) .
Explanation
Understanding Direct Variation Since y varies directly as x , we can express their relationship as y = k x , where k is the constant of proportionality. This means that as x increases, y increases proportionally.
Finding the Constant of Proportionality We are given that y = 18 when x = 5 . We can use this information to find the constant of proportionality, k . Substituting these values into the equation y = k x , we get 18 = k × 5 .
Calculating k To solve for k , we divide both sides of the equation 18 = 5 k by 5: k = 5 18 .
Writing the Direct Variation Equation Now that we have found k , we can write the direct variation equation as y = 5 18 x . This equation allows us to find y for any given value of x .
Finding y when x = 11 We want to find the value of y when x = 11 . Substituting x = 11 into the equation y = 5 18 x , we get y = 5 18 × 11 .
Final Expression 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 5 miles in 1 hour, then the constant of proportionality is 5 miles per hour. Therefore, if you travel for 3 hours, you would travel 15 miles, calculated as d i s t an ce = 5 × 3 . Understanding direct variation helps in predicting outcomes based on known relationships.
The relationship of direct variation between y and x can be expressed as y = k x , where k is the constant of proportionality found using the values y = 18 and x = 5 . The resulting equation is y = 5 18 x , and when substituting x = 11 , we find that y can be expressed as y = 5 18 ( 11 ) . Therefore, the correct option is B : y = 5 18 ( 11 ) .
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