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Analyze the meaning of \( f \) from part A. in terms of number of Franklin Tress per year (in millions). Then determine the number of years it will take after 2020 to reach a species of 2 million Franklin Tress. Use the graphing calculator to help explain.

Ask by Henry Martin. in the United States
Feb 26,2025

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Answer

To determine how many years it will take after 2020 to reach 2 million Franklin Trees, use the graphing calculator to find where the function \( f(t) \) equals 2. Plot \( f(t) \) and the line \( y = 2 \), then find their intersection point to get the required number of years.

Solution

To analyze the meaning of \( f \) in terms of the number of Franklin Trees per year (in millions), we first need to understand what \( f \) represents. Typically, in such contexts, \( f(t) \) could represent the population of Franklin Trees at time \( t \), where \( t \) is measured in years since a certain starting point (e.g., 2020). ### Step 1: Understanding \( f(t) \) Assuming \( f(t) \) is a function that models the growth of Franklin Trees, we can interpret \( f(t) \) as follows: - \( f(t) \) gives the number of Franklin Trees (in millions) at year \( t \). - For example, if \( f(0) = 1 \), it means there are 1 million Franklin Trees in the year 2020. ### Step 2: Finding the Year to Reach 2 Million Trees To determine how many years it will take after 2020 to reach a population of 2 million Franklin Trees, we need to solve the equation: \[ f(t) = 2 \] This means we need to find the value of \( t \) such that the function \( f(t) \) equals 2. ### Step 3: Using a Graphing Calculator To find the solution graphically, we can plot the function \( f(t) \) and look for the point where it intersects the line \( y = 2 \). 1. **Plot the function \( f(t) \)** on the graphing calculator. 2. **Draw the horizontal line** \( y = 2 \). 3. **Identify the intersection point** of the two graphs. ### Step 4: Calculation If you have the specific function \( f(t) \), please provide it so I can perform the calculations and find the exact year when the population reaches 2 million. If not, I can proceed with a general approach to find the intersection point. Please provide the function \( f(t) \) or any additional details needed to proceed.

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Let's break this down! When analyzing \( f \), it typically represents the growth function of Franklin Tress over time. If \( f(t) \) measures the population in millions of Franklin Tress at year \( t \), you can compare the function's outputs to understand its growth rate. If \( f(t) = 0.5t + 1 \), for example, by the year 2020 (where \( t=0 \)), there are \( 1 \) million Franklin Tress. To find when it will reach \( 2 \) million, set \( f(t) = 2 \) and solve for \( t \). You'd rearrange and solve the equation: \( 0.5t + 1 = 2 \). This simplifies to \( 0.5t = 1 \) or \( t = 2 \). Adding this to 2020, it means by the year 2022, we can expect to witness a lovely increase to 2 million Franklin Tress! To visualize this, using a graphing calculator to plot the function will show you how \( f(t) \) climbs. The x-axis represents the years (with 2020 as the starting point), and the y-axis covers the millions of Franklin Tress. The intersection of your \( f(t) \) curve and the line at \( 2 \) million showcases the exact point in time—super helpful for visual learners!

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Question 4 Consider the following three systems of linear equations. \[ \begin{array}{c} \text { System A } \\ \left\{\begin{array} { c } { \text { System B } } \\ { - 2 x + 3 y = - 7 [ \mathrm { A } 1 ] } \\ { - 7 x + 4 y = 8 } \end{array} \left[\begin{array} { c } { \text { [A2] } } \end{array} \left\{\begin{array} { r } { - 2 x + 3 y = - 7 [ \mathrm { B } 1 ] } \\ { 2 1 x - 1 2 y = - 2 4 [ \mathrm { B } 2 ] } \end{array} \left\{\left\{\begin{array}{c} -2 x+3 y=-7[\mathrm{C} 1] \\ 13 x=-52 \end{array}[\mathrm{C} 2]\right.\right.\right.\right.\right. \end{array} \] Answer the questions below. For each, choose the transformation and then fill in the blank with the correct number. The arrow ( \( \rightarrow \) ) means the expression on the left becomes the expression on the right. (a) How do we transform System A into System B? \( \square \) \( \times \) Equation \( [\mathrm{A} 1] \rightarrow \) Equation \( [ \) B1] \( -3 \times \) Equation [A2] \( \rightarrow \) Equation [B2] \( \square \) \( \times \) Equation \( [\mathrm{A} 1]+ \) Equation \( [\mathrm{A} 2] \rightarrow \) Equation \( [\mathrm{B} 2] \) \( \square \) \( \times \) Equation \( [A 2]+ \) Equation \( [A 1] \rightarrow \) Equation \( [B 1] \) (b) How do we transform System B into System C? \( \times \) Equation [B1] \( \rightarrow \) Equation [C1] \( \square \) \( \times \) Equation [B2] \( \rightarrow \) Equation [C2] \( \square \) \( \times \) Equation \( [\mathrm{B} 1]+ \) Equation \( [\mathrm{B} 2] \rightarrow \) Equation \( [\mathrm{C} 2] \) \( \square \) \( \times \) Equation \( [\mathrm{B} 2]+ \) Equation \( [\mathrm{B} 1] \rightarrow \) Equation \( [\mathrm{C} 1] \)

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