Final answer:
To find the price at which demand is unit elastic, you need to find the price at which the percentage change in demand is equal to the percentage change in price. This can be done by setting up an equation and solving for the value of p. At a unit elastic price, the company can maximize its revenue by finding a balance between the price and the quantity sold.
Step-by-step explanation:
To find the price at which the demand is unit elastic, we need to find the price at which the percentage change in demand is equal to the percentage change in price.
Let's start by finding the percentage change in demand. The formula for percentage change is:
(new value - original value) / original value * 100
Substituting the values into the formula, we have:
(D(p) - D(p + Δp)) / D(p) * 100 = (5000 - 500(p + Δp)^2 - (5000 - 500p^2)) / (5000 - 500p^2) * 100 = Δp / (500 - 50p^2) * 100
Next, let's find the percentage change in price. Since we are looking for a unit elastic demand, the percentage change in price will be equal to the percentage change in demand:
Δp / p * 100 = Δp / (500 - 50p^2) * 100
Now, we can set up the equation:
Δp / (500 - 50p^2) * 100 = Δp / p * 100
Cross multiplying, we get:
p * (500 - 50p^2) = 500 - 50p^2
Expanding and rearranging, we have:
500p - 50p^3 = 500 - 50p^2
50p^3 - 50p^2 + 500p - 500 = 0
Factoring out a common factor of 50, we get:
50(p^3 - p^2 + 10p - 10) = 0
Dividing both sides by 50, we have:
p^3 - p^2 + 10p - 10 = 0
This cubic equation can be solved using numerical methods or graphical methods to find the value of p at which the demand is unit elastic. The exact value and an approximation with two significant digits will depend on the specific values of the constants in the equation.
In terms of revenue, this price means that the company will maximize its revenue by selling the maximum quantity of broccoli. At a unit elastic price, the company is able to find the balance between maximizing price and maximizing quantity sold.