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. $\newline$ The consumer surplus for a certain product increases at a rate of $\frac{900}{(x+13)}-35$ dollars per thousand units of the product sold (where $x$ is the number of thousands of units sold). $\newline$ By how many dollars does the surplus increase between $x=7$ and $x=12$ ? $\newline$ Choose $1$ answer: $\newline$ (A) $900 \ln (0.8)-35$ $\newline$ (B) $900 \ln (1.25)-35$ $\newline$ (C) $900 \ln (0.8)-175$ $\newline$ (D) $900 \ln (1.25)-175$

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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.

\newline

The consumer surplus for a certain product increases at a rate of

\frac{900}{(x+13)}-35

dollars per thousand units of the product sold (where

x

is the number of thousands of units sold).

\newline

By how many dollars does the surplus increase between

x=7

and

x=12

\newline

Choose

1

answer:

\newline

(A)

900 \ln (0.8)-35

\newline

(B)

900 \ln (1.25)-35

\newline

(C)

900 \ln (0.8)-175

\newline

(D)

900 \ln (1.25)-175

Given Rate Function: We are given the rate of increase of consumer surplus as a function of $x$ , which is $\frac{900}{x+13} - 35$ 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 $x=12$ . $\int_{7}^{12} \left[\frac{900}{x+13} - 35\right] dx$
Separate into Parts: Separate the integral into two parts: one for $\frac{900}{x+13}$ and another for $-35$ . $\int_{7}^{12} \frac{900}{x+13} \, dx - \int_{7}^{12} 35 \, dx$
Calculate First Part: The integral of $\frac{900}{x+13}$ with respect to $x$ is $900\ln|x+13|$ , and the integral of a constant is just the constant times the variable. $\newline$ Calculate the first part: $[900\ln|x+13|]$ from $7$ to $12$ .
Substitute Limits: Substitute the upper and lower limits into the first part of the integral. $\newline$ $900\ln|12+13| - 900\ln|7+13|$
Simplify Logarithms: Simplify the expression using properties of logarithms. $\newline$ $900\ln(25) - 900\ln(20)$
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. $\newline$ $-35(12) - (-35(7))$
Combine Logarithmic Terms: Simplify the expression. $-420 + 245$
Simplify Expression: Combine the results from the two parts of the integral. $900\ln(25) - 900\ln(20) - 420 + 245$
Match with Answer: Use the properties of logarithms to combine the logarithmic terms. $\newline$ $900\ln(\frac{25}{20}) - 175$
Match with Answer: Use the properties of logarithms to combine the logarithmic terms. $\newline$ $900\ln(\frac{25}{20}) - 175$ Simplify the logarithmic expression. $\newline$ $900\ln(1.25) - 175$
Match with Answer: Use the properties of logarithms to combine the logarithmic terms. $\newline$ $900\ln(\frac{25}{20}) - 175$ Simplify the logarithmic expression. $\newline$ $900\ln(1.25) - 175$ Match the final expression with the given answer choices. $\newline$ The correct answer is (B) $900 \ln(1.25)-175$ .