For the demand function $q=D(p)=\sqrt{471-p}$, find the following.
a) The elasticity
b) The elasticity at $p=77$, stating whether the demand is elastic, inelastic or has unit elasticity
c) The value(s) of $p$ for which total revenue is a maximum (assume that $p$ is in dollars)
a) Find the equation for elasticity.
\[
E(p)=\frac{p}{2(471-p)}
\]
b) Find the elasticity at the given price, stating whether the demand is elastic, inelastic or has unit elasticity. $E(77)=\square$ (Simplify your answer. Type an integer or a fraction.)

Step 1 ：The elasticity of demand is a measure of how much the quantity demanded of a good responds to a change in the price of that good. It is calculated as the percentage change in quantity demanded divided by the percentage change in price. In this case, we have the formula for elasticity given as \(E(p)=\frac{p}{2(471-p)}\).

Step 2 ：We need to substitute \(p=77\) into this formula to find the elasticity at this price.

Step 3 ：After substituting, we get the value of elasticity at \(p=77\) as approximately 0.098.

Step 4 ：If the elasticity is greater than 1, the demand is elastic. If it is less than 1, the demand is inelastic. If it is equal to 1, the demand has unit elasticity.

Step 5 ：The elasticity at \(p=77\) is approximately 0.098, which is less than 1. Therefore, the demand is inelastic at this price.

Step 6 ：Final Answer: The elasticity at \(p=77\) is approximately 0.098, which means the demand is inelastic. Therefore, the final answer is \(\boxed{0.098}\).