Question

If y is directly proportional to the square of x, when x=4 y = 25, how do you find an expression for y in terms of x?

Answer 1

Hint: First try to get an equation in terms of variables and constant by removing the proportionality. Substitute the variables by the given values to get the value of the constant. Then again substitute the value of the constant in the base equation to get the required expression.

Complete step by step answer:
According to the question; y is directly proportional to the square of x
\[\Rightarrow y\propto {{x}^{2}}\]
\[\Rightarrow y=k{{x}^{2}}\](Where k is the proportionality constant)
Therefore, y would be in the form \[y=k{{x}^{2}}\] (where k is a constant)
Now the given values are x=4 and y=25.
Putting these values in the above equation we get,
$\begin{align}
  & 25=k\times {{4}^{2}} \\
 & \Rightarrow 25=16k \\
\end{align}$
The value of the constant k obtained from the equation is $\dfrac{25}{16}$
Now we have to find an expression of y in terms of x.
This can be expressed by substituting the value of the constant k in the base equation \[y=k{{x}^{2}}\]
Putting the value of constant $k=\dfrac{25}{16}$(in fraction form);
We get the expression $y=\dfrac{25}{16}{{x}^{2}}$.

Note: A proportionality constant ‘k’ should be added after removing the proportionality sign. The proportionality constant ‘k’ can be obtained by putting the values of ‘x’ and ‘y’ in the proportionality equation.