What is the value of Q (rounded to the nearest integer) when demand is unit elastic?
The value of Q (rounded to the nearest integer) when demand is unit elastic is 617.
Understanding Unit Elastic Demand
Unit elastic demand refers to a situation where the percentage change in quantity demanded is equal to the percentage change in price. In other words, when demand is unit elastic, consumers are equally responsive to changes in price with changes in quantity demanded.
Finding the Value of Q
To find the value of Q when demand is unit elastic, we start with the equation Q = 617 - 3p^2, where p represents the price. Our goal is to determine the price at which the percentage change in quantity demanded matches the percentage change in price.
First, we differentiate the equation Q = 617 - 3p^2 with respect to p. The derivative gives us dQ/dp = -6p.
Next, we consider the percentage change in quantity demanded (%ΔQ) and the percentage change in price (%Δp). Since we want unit elastic demand, %ΔQ should be equal to %Δp. This leads us to the equation: ΔQ / Q = Δp / p.
Substituting the derivative dQ/dp into the equation, we get (-6p) / (617 - 3p^2) = Δp / p. Simplifying further, we find -6p = 617Δp - 3p^2Δp.
By taking the limit as Δp approaches zero, we solve for p = 0. Therefore, when the price is zero, the quantity demanded is Q = 617.
In conclusion, when demand is unit elastic, the value of Q (rounded to the nearest integer) is 617. This means that consumers are equally responsive to price changes, resulting in a balanced relationship between quantity demanded and price.