2) At the beginning of the year, the number of books owned by a library was Since then, it has grown by \( 1 \% \) each month. Which expressions represent the number of books, in thousands, owned library 5 years later if it continues to grow at that rate? \( \begin{array}{ll}\text { A) } 10 \cdot\left((1+0.01)^{12}\right)^{5} & \text { B) } 10 \cdot\left(1+0.01^{12}\right)^{60} \\ \text { C) } 10 \cdot(1.01)^{5} & \text { D) } 10 \cdot(1.01)^{60} \\ \text { E) } 10 \cdot(0.01)^{60} & \end{array} \)

To solve the problem, we need to determine the expression that represents the number of books owned by the library after 5 years, given that the library's collection grows by \(1\%\) each month. ### Step 1: Understand the growth rate The library grows by \(1\%\) each month, which can be expressed as a growth factor of \(1 + 0.01 = 1.01\). ### Step 2: Calculate the total number of months in 5 years Since there are 12 months in a year, over 5 years, the total number of months is: \[ \text{Total months} = 5 \times 12 = 60 \] ### Step 3: Write the expression for growth The number of books after 5 years can be expressed using the formula for exponential growth: \[ \text{Number of books} = \text{Initial number of books} \times (\text{Growth factor})^{\text{Number of periods}} \] In this case, the initial number of books is \(10\) (in thousands), the growth factor is \(1.01\), and the number of periods (months) is \(60\). Thus, the expression becomes: \[ \text{Number of books} = 10 \times (1.01)^{60} \] ### Step 4: Identify the correct option Now, let's compare this expression with the provided options: - **A)** \(10 \cdot\left((1+0.01)^{12}\right)^{5}\) - **B)** \(10 \cdot\left(1+0.01^{12}\right)^{60}\) - **C)** \(10 \cdot(1.01)^{5}\) - **D)** \(10 \cdot(1.01)^{60}\) - **E)** \(10 \cdot(0.01)^{60}\) The correct expression that matches our derived formula is: \[ \text{D) } 10 \cdot(1.01)^{60} \] ### Final Answer The expression that represents the number of books owned by the library 5 years later, if it continues to grow at that rate, is: **D) \(10 \cdot(1.01)^{60}\)**.