Question: 3. Joey has been planting trees in his garden. The maple tree is 20 inches tall, growing at 2…

3. Joey has been planting trees in his garden. The maple tree is 20 inches tall, growing at 2 inches per month. The oak tree is 30 inches tall and grows 1 inch per month. Write the equations. Graph the system. Write the solution. What does the solution represent?

Solution

Joey has been planting trees in his garden. The maple tree is 20 inches tall, growing at 2 inches per month. The oak tree is 30 inches tall and grows 1 inch per month. Write the equations. Let \( t \) represent the number of months. For the maple tree: Initial height: 20 inches Growth rate: 2 inches per month Equation: \[ h_m(t) = 20 + 2t \] For the oak tree: Initial height: 30 inches Growth rate: 1 inch per month Equation: \[ h_o(t) = 30 + t \] Graph the system. The graph will show the height of each tree over time, with the x-axis representing time in months and the y-axis representing the height in inches. Write the solution. What does the solution represent? Set the equations equal to find when both trees will be the same height: \[ 20 + 2t = 30 + t \] Subtract \( t \) from both sides: \[ 20 + t = 30 \] Subtract 20 from both sides: \[ t = 10 \] After 10 months, both trees will be the same height. To find the height at this time, substitute \( t = 10 \) into either equation: \[ h_m(10) = 20 + 2(10) = 40 \] So, after 10 months, both trees will be 40 inches tall. The solution represents the time and height at which both trees are the same height.