Question

If \[y\] varies directly as the square of \[x\]. When \[x = 2,y = 20\], how do you find \[y\], when \[x = 4\]?

Answer 1

Hint: In the question it is given that \[y\] varies directly as the square of \[x\]. So, with this information we will set up a direct proportional relation. Then we will remove the proportionality sign with a constant. Then we will put the value of the given pair in the equation and from there we will find the value of the constant. Then with the constant value we will make a general equation and then from that equation we will find the required value.

Complete step-by-step solution:
Given;
\[y\] varies directly as the square of \[x\]. Writing it mathematically we get;
\[y \propto {x^2}\]
On removing the proportionality sign we get;
\[ \Rightarrow y = k{x^2}\]
Now it is given that when \[x = 2,y = 20\]. So, putting these values in the equation we get;
\[ \Rightarrow 20 = k{\left( 2 \right)^2}\]
On solving we get;
\[ \Rightarrow k = \dfrac{{20}}{4}\]
\[ \Rightarrow k = 5\]
Hence the general equation we get by putting the value of constant is;
\[ \Rightarrow y = 5{x^2}\]
Now we will put \[x = 4\], in this equation to find the value of \[y\]. So, we have;
\[ \Rightarrow y = 5{\left( 4 \right)^2}\]
On solving we get;
\[ \Rightarrow y = 5 \times 16\]
On calculating we get;
\[ \Rightarrow y = 80\]

Hence when \[x = 4\], \[y = 80\].

Note: One thing to note here is that since there is a direct relation, when one of the quantities increases the other will also increase and when one decreases the other one will also decrease. We can also see from the general equation that if we draw a curve for the given relation then we will get a parabola, which passes through the origin and whose axis is x-axis.