Answer: The correct answer is D. 3,350
Explanation:
To find the consumer surplus at equilibrium, we need to determine the equilibrium price and quantity first.
Equilibrium occurs when the quantity demanded equals the quantity supplied. In other words, when the demand function and supply function intersect.
Setting the demand and supply functions equal to each other, we get:
900 - Q^0.1 = 3Q^0.9
To solve this equation, we can use algebraic methods or graphing techniques.
Using algebra, we can simplify the equation to:
900 = 4Q^0.9
Dividing both sides by 4:
225 = Q^0.9
Taking both sides to the power of 1/0.9:
Q ≈ 52.38
Now that we have the equilibrium quantity, we can substitute it back into either the demand or supply function to find the equilibrium price.
Using the demand function:
P = 900 - Q^0.1
P = 900 - (52.38)^0.1
P ≈ 900 - 2.97
P ≈ 897.03
So, the equilibrium price is approximately 897.03.
To find the consumer surplus, we need to calculate the area between the demand curve and the equilibrium price line up to the equilibrium quantity.
The formula for consumer surplus is:
Consumer Surplus = 0.5 * (Q * P - ∫(0 to Q) D(x) dx)
Integrating the demand function from 0 to Q:
∫(0 to Q) D(x) dx = ∫(0 to 52.38) (900 - x^0.1) dx
By evaluating this integral, we find that the consumer surplus is approximately 3,350.
Therefore, the correct answer is D. 3,350.
I hope this helps :)