Write an equation that relates the variables. z is directly proportional to x and inversely proportional to the fourth ext { fourth } fourth root of yA) z = k x 4 y z = \frac { k } { x ^ { 4 } \sqrt { y } } z = x 4 y k B) z = x k y 4 z = x \sqrt [ 4 ] { k y } z = x 4 k y C) z = 1 k x 4 y z = \frac { 1 } { k x ^ { 4 } \sqrt { y } } z = k x 4 y 1 D) z = k x y 4 z = \frac { k x } { \sqrt [ 4 ] { y } } z = 4 y k x E) z = k x y 4 z = k x \sqrt [ 4 ] { y } z = k x 4 y

Question

Write an equation that relates the variables. z is directly proportional to x and inversely proportional to the    fourth  	ext { fourth }  fourth    root of yA)    z = k x 4 y z = \frac { k } { x ^ { 4 } \sqrt { y } } z = x 4 y ​ k ​ B)    z = x k y 4 z = x \sqrt [ 4 ] { k y } z = x 4 k y ​ C)    z = 1 k x 4 y z = \frac { 1 } { k x ^ { 4 } \sqrt { y } } z = k x 4 y ​ 1 ​ D)    z = k x y 4 z = \frac { k x } { \sqrt [ 4 ] { y } } z = 4 y ​ k x ​ E)    z = k x y 4 z = k x \sqrt [ 4 ] { y } z = k x 4 y ​

Everton McNaughton · Accepted Answer

Final Answer : D, Explanation :  Since z is directly proportional to x, we have z=kx for some constant k. Also, z is inversely proportional to the fourth root of y, which means z is proportional to 1/y^(1/4). Putting these together, we get z=kx/y^(1/4). Solving for k, we get k=zy^(1/4)/x. Substituting this value of k back into the equation, we get z=(zy^(1/4)/x)x/(y^(1/4))=zy^(1/4)/y^(1/4)=z. Therefore, the equation simplifies to z=kx/y^(1/4), which is equivalent to z=kx/sqrt[4]{y}. So our answer is D.

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