A manufacturer has determined that when x thousand units of a particular commodity are produced, the price at which all the units can be sold is p = F (x) dollars per unit, where F is
the demand function
F (x) = 10 −xe^(0.02x)
a) At what price are 5000 (x0 = 5) units demanded?
b) Find the consumers’ surplus when 5000 units are demanded!
I really do not get how to solve this. Thank you so much!
To solve this problem, we need to find the price at which 5000 units are demanded, and then determine the consumer's surplus.
a) To find the price at which 5000 units are demanded, we need to evaluate the demand function, F(x), when x = 5 (since 5000 units = 5 * 1000).
Step 1: Substitute x = 5 into the demand function:
F(5) = 10 − 5e^(0.02*5)
Step 2: Simplify the expression:
F(5) = 10 − 5e^(0.1)
= 10 - 5e^0.1
So, the price at which 5000 units are demanded is given by F(5) = 10 - 5e^0.1 dollars per unit.
b) To find the consumer's surplus for 5000 units, we first need to calculate the total benefits from the demand function, F(x), for x = 5.
Step 1: Integrate the demand function from 0 to 5:
B(x) = ∫[0,5] F(t) dt
Step 2: Substitute the given demand function:
B(x) = ∫[0,5] (10 - te^(0.02t)) dt
Step 3: Evaluate the integral:
B(x) = [10t - e^(0.02t)] (from 0 to 5)
Step 4: Substitute the upper and lower limits:
B(x) = [10*5 - e^(0.02*5)] - [10*0 - e^(0.02*0)]
Step 5: Simplify the expression:
B(x) = [50 - e^(0.1)] - [0 - e^(0)]
= [50 - e^(0.1)] - [0 - 1]
So, the consumer's surplus for 5000 units is B(x) = 50 - e^(0.1) - (-1).