Consumer surplus is the amount of money that consumers save because they are able to buy a product at a price lower than the highest price they would be willing to pay.The consumer surplus for a certain product increases at a rate of (x+13)900−35 dollars per thousand units of the product sold (where x is the number of thousands of units sold).By how many dollars does the surplus increase between x=7 and x=12 ?Choose 1 answer:(A) 900ln(0.8)−35(B) 900ln(1.25)−35(C) 900ln(0.8)−175(D) 900ln(1.25)−175
Q. Consumer surplus is the amount of money that consumers save because they are able to buy a product at a price lower than the highest price they would be willing to pay.The consumer surplus for a certain product increases at a rate of (x+13)900−35 dollars per thousand units of the product sold (where x is the number of thousands of units sold).By how many dollars does the surplus increase between x=7 and x=12 ?Choose 1 answer:(A) 900ln(0.8)−35(B) 900ln(1.25)−35(C) 900ln(0.8)−175(D) 900ln(1.25)−175
Given Rate Function: We are given the rate of increase of consumer surplus as a function of x, which is x+13900−35 dollars per thousand units. To find the total increase in consumer surplus between x=7 and x=12, we need to integrate this rate function with respect to x from 7 to 12.
Set up Integral: Set up the integral of the rate function from x=7 to x=12. ∫712[x+13900−35]dx
Separate into Parts: Separate the integral into two parts: one for x+13900 and another for −35.∫712x+13900dx−∫71235dx
Calculate First Part: The integral of x+13900 with respect to x is 900ln∣x+13∣, and the integral of a constant is just the constant times the variable.Calculate the first part: [900ln∣x+13∣] from 7 to 12.
Substitute Limits: Substitute the upper and lower limits into the first part of the integral.900ln∣12+13∣−900ln∣7+13∣
Simplify Logarithms: Simplify the expression using properties of logarithms.900ln(25)−900ln(20)
Calculate Second Part: The integral of −35 with respect to x from 7 to 12 is −35x, evaluated from 7 to 12. Calculate the second part: [−35x] from 7 to 12.
Combine Results: Substitute the upper and lower limits into the second part of the integral. −35(12)−(−35(7))
Combine Logarithmic Terms: Simplify the expression. −420+245
Simplify Expression: Combine the results from the two parts of the integral. 900ln(25)−900ln(20)−420+245
Match with Answer: Use the properties of logarithms to combine the logarithmic terms. 900ln(2025)−175
Match with Answer: Use the properties of logarithms to combine the logarithmic terms.900ln(2025)−175 Simplify the logarithmic expression.900ln(1.25)−175
Match with Answer: Use the properties of logarithms to combine the logarithmic terms.900ln(2025)−175 Simplify the logarithmic expression.900ln(1.25)−175 Match the final expression with the given answer choices.The correct answer is (B) 900ln(1.25)−175.