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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 
(900)/((x+13))-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) 
900 ln(0.8)-35
(B) 
900 ln(1.25)-35
(C) 
900 ln(0.8)-175
() 
900 ln(1.25)-175

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.\newlineThe consumer surplus for a certain product increases at a rate of 900(x+13)35 \frac{900}{(x+13)}-35 dollars per thousand units of the product sold (where x x is the number of thousands of units sold).\newlineBy how many dollars does the surplus increase between x=7 x=7 and x=12 x=12 ?\newlineChoose 11 answer:\newline(A) 900ln(0.8)35 900 \ln (0.8)-35 \newline(B) 900ln(1.25)35 900 \ln (1.25)-35 \newline(C) 900ln(0.8)175 900 \ln (0.8)-175 \newline(D) 900ln(1.25)175 900 \ln (1.25)-175

Full solution

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.\newlineThe consumer surplus for a certain product increases at a rate of 900(x+13)35 \frac{900}{(x+13)}-35 dollars per thousand units of the product sold (where x x is the number of thousands of units sold).\newlineBy how many dollars does the surplus increase between x=7 x=7 and x=12 x=12 ?\newlineChoose 11 answer:\newline(A) 900ln(0.8)35 900 \ln (0.8)-35 \newline(B) 900ln(1.25)35 900 \ln (1.25)-35 \newline(C) 900ln(0.8)175 900 \ln (0.8)-175 \newline(D) 900ln(1.25)175 900 \ln (1.25)-175
  1. Given Rate Function: We are given the rate of increase of consumer surplus as a function of xx, which is 900x+1335\frac{900}{x+13} - 35 dollars per thousand units. To find the total increase in consumer surplus between x=7x=7 and x=12x=12, we need to integrate this rate function with respect to xx from 77 to 1212.
  2. Set up Integral: Set up the integral of the rate function from x=7x=7 to x=12x=12. 712[900x+1335]dx\int_{7}^{12} \left[\frac{900}{x+13} - 35\right] dx
  3. Separate into Parts: Separate the integral into two parts: one for 900x+13\frac{900}{x+13} and another for 35-35.712900x+13dx71235dx\int_{7}^{12} \frac{900}{x+13} \, dx - \int_{7}^{12} 35 \, dx
  4. Calculate First Part: The integral of 900x+13\frac{900}{x+13} with respect to xx is 900lnx+13900\ln|x+13|, and the integral of a constant is just the constant times the variable.\newlineCalculate the first part: [900lnx+13][900\ln|x+13|] from 77 to 1212.
  5. Substitute Limits: Substitute the upper and lower limits into the first part of the integral.\newline900ln12+13900ln7+13900\ln|12+13| - 900\ln|7+13|
  6. Simplify Logarithms: Simplify the expression using properties of logarithms.\newline900ln(25)900ln(20)900\ln(25) - 900\ln(20)
  7. Calculate Second Part: The integral of 35-35 with respect to xx from 77 to 1212 is 35x-35x, evaluated from 77 to 1212. Calculate the second part: [35x][-35x] from 77 to 1212.
  8. Combine Results: Substitute the upper and lower limits into the second part of the integral. \newline35(12)(35(7))-35(12) - (-35(7))
  9. Combine Logarithmic Terms: Simplify the expression. 420+245-420 + 245
  10. Simplify Expression: Combine the results from the two parts of the integral. 900ln(25)900ln(20)420+245900\ln(25) - 900\ln(20) - 420 + 245
  11. Match with Answer: Use the properties of logarithms to combine the logarithmic terms. \newline900ln(2520)175900\ln(\frac{25}{20}) - 175
  12. Match with Answer: Use the properties of logarithms to combine the logarithmic terms.\newline900ln(2520)175900\ln(\frac{25}{20}) - 175 Simplify the logarithmic expression.\newline900ln(1.25)175900\ln(1.25) - 175
  13. Match with Answer: Use the properties of logarithms to combine the logarithmic terms.\newline900ln(2520)175900\ln(\frac{25}{20}) - 175 Simplify the logarithmic expression.\newline900ln(1.25)175900\ln(1.25) - 175 Match the final expression with the given answer choices.\newlineThe correct answer is (B) 900ln(1.25)175900 \ln(1.25)-175.

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