Question

If \[x \] varies directly with \[{y^2} \] and \[x = 4 \] when \[y = 5 \] , then find \[x \] when \[y \] is 15.
A.4
B.9
C.12
D.36

Answer 1

Hint: Here, we need to use the concept of direct variation to solve the problem. We know that when two variables are directly proportional to each other, then we can equate them by multiplying one of them with the constant of variation or the constant of proportionality. After this, we will use the given values of both the parameters to find the value of this constant. Then, by putting the value of the constant in the equation again, we can find the value of the remaining variable.
Formula used:
 $ a \propto b \Rightarrow a = kb $ , where, $ a $ and $ b $ are in direct proportion with each other and $ k $ is the constant of variation or the constant of proportionality

Complete step-by-step answer:
In the problem, we are given that \[x \] varies directly with \[{y^2} \] which means that \[x \] is directly proportional to \[{y^2} \] . Therefore, by using the formula $ a \propto b \Rightarrow a = kb $ , we can write that
 $ x \propto {y^2} \Rightarrow x = k{y^2} $
Now, our first step is to find the value of the constant of variation $ k $ in the equation $ x = k{y^2} $ by using the initial given values of both the parameters $ x $ and $ y $ .
 $ x = k{y^2} $
It is given that
 \[x = 4 \] when \[y = 5 \]
 $
    \Rightarrow 4 = k{ \left( 5 \right)^2} \\
    \Rightarrow 4 = 25k \\
    \Rightarrow k = \dfrac{4}{{25}} \;
  $
Now by putting this value of constant in the equation $ x = k{y^2} $ , we get
 $ x = \dfrac{4}{{25}}{y^2} $
Now, it is asked that we need to find the value of $ x $ when $ y = 15 $ . For this, we will put this value $ y = 15 $ in the equation $ x = \dfrac{4}{{25}}{y^2} $ .
 \[
    \Rightarrow x = \dfrac{4}{{25}} \times { \left( {15} \right)^2} \\
    \Rightarrow x = \dfrac{4}{{25}} \times 225 \\
    \Rightarrow x = 4 \times 9 \\
    \Rightarrow x = 36 \;
  \]
Thus, when $ y = 15 $ , the value of the other variable $ x = 36 $
So, the correct answer is “x=36 when y=15”.

Note: Here, the value of the constant of variation remains unchanged for any values of both parameters. Therefore, we can find the value of one parameter for different values of the other one using the given relation between them. We can plot these values of both parameters and obtain a graph showing the relation between them.