what is the effect on P if x is increase by %?

Question

Let P vary inversely as the square of x. What is the effect on P if x is increased by %?

Answer 1

P=k/x² where k is a constant defines the relationship.

If x is increased by p% then its value becomes x+px/100. Let r=p/100, so x becomes x(1+r).

P becomes k/[x²(1+r)²] so if the original value of P is P₀ and P_r is its new value after the increase, then:

P_r-P₀ is the change in P=k/[x²(1+r)²]-k/x²,

P_r-P₀=(k/x²)(1/(1+r)²-1)=P₀(1-(1+r)²)/(1+r)²,

P_r-P₀=P₀(-r(2+r))/(1+r)²=-rP₀(2+r)/(1+r)², which is a decrease.

We can also write this in terms of p (the percentage increase in x):

Change in P=-pP₀(2+p/100)/(1+p/100)² so P decreases by a factor p(2+p/100)/(1+p/100)²% of its original value.

If p=100% then P_r=P₁ and P₁-P₀=k/(4x²)-k/x²=-¾P₀, a decrease of 75%.

If p=10%, r=0.1 and the change is -10(2.1)/1.21=-21/1.21=-17.355% or a decrease of about 17.355%.