The number of people in Bernardo's social network is increasing at a rate of \( r(t)=-2(t-3)^{2}+23 \) people per month (where \( t \) is the time in months since Bernardo set up the network). At time \( t=4 \) months, Bernardo had 80 people in his social network. How many people were in Bernardo's social network at the end of the \( 6^{\text {th }} \) month? Which expression can we use to solve the problem? Choose 1 answer: (A) \( 80+\int_{4}^{6} r(t) d t \) (B) \( r(6) \) (C) \( \int_{4}^{6} r(t) d t \) (D) \( \int_{0}^{6} r(t) d t \)

To find the number of people in Bernardo's social network at the end of the 6th month, we need to account for the initial number of people he had (80) and how many new people joined between months 4 and 6. The correct expression to find this is indeed (A) \( 80+\int_{4}^{6} r(t) d t \), which gives us the initial amount plus the integral of the rate of increase over that interval, capturing the total addition to the social network. Now, let's highlight how this kind of problem is prevalent in real life! People often use integrals in business models to predict growth—such as user acquisition over time—making decisions based on these projections have proven crucial for startups and social media platforms alike. This mathematical approach helps them strategize effectively about features, marketing, and user engagement. Another fascinating aspect is how functions like \( r(t) \) can reflect real-world phenomena. The shape of the function, with its downward opening parabola, suggests there's a maximum growth rate at a certain time—specifically at \( t=3 \) months. This resembles many real-life situations, where growth rates peak before eventually slowing down. Understanding these patterns can provide meaningful insights for effective planning and adaptability in changing environments!