The demand equation for a manufacturer's product has the form $ q = a*\sqrt{b-cp^2}$. Where, a, b, and c are positive integers.
a) Determine the interval of prices for which demand is elastic.
b) For which price is there unit elasticity?
Notes: Elasticity formula: $$\frac{p}{q} * \frac{dq}{dp} $$
and it is elastic when the above formula bigger than 1 and unit elastic when it is equal to one.
So, I took the derivative of q to find the right side of the formula and I got: $$\frac{-acp}{\sqrt{b-cp^2}}$$ Then I wrote the left hand side of the formula as $$\frac{p}{a\sqrt{b-cp^2}}$$. And when I multiplied the left hand side with right hand side I got $$\frac{-cp^2}{b-cp^2}.$$
So now I need to find the intervals for p when this expression is bigger than 1 but if you try to solve the inequality only thing you left is 0 > b.
I don't know where I went wrong. Please explain it. Thank you in advance.
Relatively Elastic is when $E_d\lt -1$
Hence $$\frac{-cp^2}{b-cp^2}\lt -1$$ $$cp^2\gt b-cp^2$$
$$2cp^2 \gt b$$
$$p^2\gt \frac{b}{2c}$$
Elastic when $p\gt \sqrt{\frac{b}{2c}}$ and $p\lt -\sqrt{\frac{b}{2c}}$
Unit Elastic when $p= \sqrt{\frac{b}{2c}}$