Question

Suppose y varies directly as the square root of x. When x = 16, y = 24. Find the constant of variation and equation of variation.

Answer 1

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HINT:If a variable varies directly with another variable then their mutual relationship can be written as
Y=cX (Where Y and X are the variables and c is the constant of variation)
Similarly, if a variable varies indirectly with another variable then their mutual relationship can be written as
Y=c \[\dfrac{1}{X}\] (Where Y and X are the variables and c is the constant of variation)

Complete step-by-step answer:
As mentioned in the question, we here only require the direct variation equation.
So, as mentioned in the question, varies directly as square root of x, so, the equation will be written as
\[y=c\sqrt{x}\] (Where c is the constant of variation and x and y are the variables)
On squaring both the side of the equation that is both left hand side as well as right hand side, we get
\[{{y}^{2}}={{c}^{2}}x\]
Now, as \[{{c}^{2}}\] is also a constant, so it can be written as another constant as ‘d’.
So, now the equation becomes
\[{{y}^{2}}=dx\]
Now, putting the values of y and x, we get
\[\begin{align}
  & {{24}^{2}}=d16 \\
 & 324=16d \\
 & d=\dfrac{24\times 24}{16} \\
 & d=36 \\
\end{align}\]
Now, we know that ‘d’= \[{{c}^{2}}\] and we also know that d=36.
So,
\[\begin{align}
  & c=\sqrt{36} \\
 & c=\pm 6 \\
\end{align}\]
But, we know that for the original equation, the values indicate that the constant of variation is positive.
 Therefore, the value of the constant of variation is +6 and the equation of variation is
\[y=6\sqrt{x}\]

NOTE:The students might make a mistake when they would try to find the sign of the constant of variation as the equation consists of a square root and that might confuse the students that which sign + or – is to be put in front of the constant of variation.