How to Solve the Gardener’s Tree Problem: A Step-by-Step Guide

Have you ever wondered how long it would take for two types of trees to be the same height if they have different initial heights and growth rates? This is a common word problem that can be solved using linear systems, algebra, and latent semantic analysis. In this article, we will show you how to solve this problem step by step, using an example of a gardener planting two types of trees. You will also learn some tips and tricks to make the problem easier and more fun. Let’s get started!

Imagine that a gardener is planting two types of trees in his backyard. Type A is three feet tall and grows at a rate of 15 inches per year. Type B is four feet tall and grows at a rate of 10 inches per year. How many years will it take for the two types of trees to be the same height?

This is a word problem that involves finding the number of years when the two types of trees are the same height. To solve this problem, we need to use some concepts and methods from linear systems, algebra, and latent semantic analysis. These are:

• Linear systems: A set of equations that can be solved by finding the values of the variables that make all the equations true at the same time.
• Algebra: A branch of mathematics that uses symbols and rules to manipulate and solve equations.
• Latent semantic analysis: A technique that uses statistics and natural language processing to analyze the meaning and relationships of words and texts.

The main steps to solve this problem are:

• Convert the feet to inches for both types of trees
• Set up an equation for each type of tree
• Solve for the number of years when the trees are the same height
• Justify the answer by checking the original equations

The main goal of this article is to show you how to solve this problem using these steps and concepts. The main outcome of this article is to help you understand and apply the concepts and methods of linear systems, algebra, and latent semantic analysis to solve similar word problems.

Convert the feet to inches

The first step to solve this problem is to convert the feet to inches for both types of trees. This is because we need to use the same unit of measurement for both types of trees, and inches are easier to work with than feet. To convert the feet to inches, we need to use a simple formula:

• One foot = 12 inches
• To convert feet to inches, multiply the number of feet by 12

For example, to convert three feet to inches, we multiply three by 12 and get 36 inches. To convert four feet to inches, we multiply four by 12 and get 48 inches.

Using this formula, we can convert the initial heights and growth rates of both types of trees to inches. Here are the results:

As you can see, we have converted the feet to inches for both types of trees and written the new heights and growth rates in inches. This will make the next step easier and more consistent.

Set up an equation for each type of tree

The second step to solve this problem is to set up an equation for each type of tree. An equation is a mathematical statement that shows the equality of two expressions. We can use an equation to model the relationship between the height and the growth rate of a tree. To write an equation for a tree, we need to use a variable and a constant.

• A variable is a symbol that can represent any value. We usually use letters such as x, y, or z to represent variables. In this problem, we will use x to represent the number of years.
• A constant is a fixed value that does not change. We usually use numbers or symbols to represent constants. In this problem, we will use the initial heights and growth rates of the trees as constants.

To write an equation for a tree, we need to follow this formula:

• Height = Initial Height + Growth Rate * Number of Years
• To write the equation, replace the words with the symbols and values

For example, to write an equation for type A tree, we use this formula:

• Height = 36 + 15 * x
• This is the equation for type A tree

To write an equation for type B tree, we use the same formula:

• Height = 48 + 10 * x
• This is the equation for type B tree

Using this formula, we can write an equation for each type of tree using the values in inches from the previous step. Here are the results:

As you can see, we have written an equation for each type of tree using the variable x and the constants 36, 15, 48, and 10. These equations show how the height of each type of tree changes over time, depending on the number of years.

Solve for the number of years

The third step to solve this problem is to solve for the number of years when the two types of trees are the same height. This is the main question of the word problem and the main unknown of the equations. To solve for the number of years, we need to use a linear system.

A linear system is a set of equations that can be solved by finding the values of the variables that make all the equations true at the same time. In this problem, we have two equations and one variable, so we have a linear system of two equations and one variable. To write a linear system, we need to use brackets and commas.

• Brackets are symbols that enclose a group of expressions. We usually use parentheses, such as ( and ), or braces, such as { and }, to represent brackets. In this problem, we will use braces to enclose the equations.
• Commas are symbols that separate different expressions. We usually use commas, such as ,, to represent commas. In this problem, we will use commas to separate the equations.

To write a linear system, we need to follow this formula:

• { Equation 1, Equation 2 }
• To write the linear system, replace the words with the equations

For example, to write a linear system for this problem, we use this formula:

• { Height = 36 + 15 x, Height = 48 + 10 x }
• This is the linear system for this problem

Using this formula, we can write a linear system using the two equations from the previous step. Here is the result:

As you can see, we have written a linear system using the brackets and the commas. This linear system shows the relationship between the heights of the two types of trees and the number of years.

To solve the linear system, we need to find the value of x that makes both equations true at the same time. This means that we need to find the value of x that makes the heights of the two types of trees equal. There are many methods to solve a linear system, such as substitution, elimination, or graphing. In this article, we will use the substitution method.

The substitution method is a method that involves replacing one variable with an equivalent expression from another equation. To use the substitution method, we need to follow these steps:

• Choose one equation and solve for one variable in terms of the other variable
• Substitute the expression for the variable into the other equation and simplify
• Solve for the remaining variable and find its value
• Substitute the value of the variable into either equation and find the value of the other variable

For example, to use the substitution method for this problem, we follow these steps:

• Choose the first equation and solve for x in terms of Height
  • Height = 36 + 15 * x
  • Subtract 36 from both sides
  • Height – 36 = 15 * x
  • Divide both sides by 15
  • (Height – 36) / 15 = x
• Substitute the expression for x into the second equation and simplify
  • Height = 48 + 10 * x
  • Replace x with (Height – 36) / 15
  • Height = 48 + 10 * (Height – 36) / 15
  • Multiply both sides by 15
  • 15 Height = 15 48 + 10 * (Height – 36)
  • Distribute the 10
  • 15 Height = 15 48 + 10 Height – 10 36
  • Combine like terms
  • 15 Height = 720 + 10 Height – 360
  • Subtract 10 * Height from both sides
  • 5 * Height = 360
  • Divide both sides by 5
  • Height = 72
• Solve for x and find its value
  • Substitute the value of Height into either equation
  • Height = 36 + 15 * x
  • Replace Height with 72
  • 72 = 36 + 15 * x
  • Subtract 36 from both sides
  • 36 = 15 * x
  • Divide both sides by 15
  • x = 2.4
• Substitute the value of x into either equation and find the value of Height
  • Height = 36 + 15 * x
  • Replace x with 2.4
  • Height = 36 + 15 * 2.4
  • Simplify
  • Height = 72

Using the substitution method, we have found the value of x and Height that make both equations true at the same time. Here are the results:

As you can see, we have solved the linear system and found the value of x and Height. These values show that the two types of trees will be the same height after 2.4 years, and that height will be 72 inches.

Justify the answer

The fourth and final step to solve this problem is to justify the answer by checking the original equations. This means that we need to verify that the values of x and Height that we found in the previous step make both equations true. To justify the answer, we need to follow these steps:

• Plug the value of x into the original equations and simplify
• Compare the values of Height from both equations and see if they are equal
• If they are equal, the answer is correct and justified
• If they are not equal, the answer is incorrect and needs to be revised

For example, to justify the answer for this problem, we follow these steps:

• Plug the value of x into the original equations and simplify
  • Height = 36 + 15 * x
  • Replace x with 2.4
  • Height = 36 + 15 * 2.4
  • Simplify
  • Height = 72
  • Height = 48 + 10 * x
  • Replace x with 2.4
  • Height = 48 + 10 * 2.4
  • Simplify
  • Height = 72
• Compare the values of Height from both equations and see if they are equal
  • Height = 72
  • Height = 72
  • The values of Height are equal
• The answer is correct and justified

Using these steps, we have justified the answer by plugging the value of x into the original equations and verifying that the values of Height are equal. Here are the results:

As you can see, we have justified the answer and confirmed that the two types of trees will be the same height after 2.4 years, and that height will be 72 inches.

Conclusion

In this article, we have shown you how to solve a word problem involving a gardener planting two types of trees with different initial heights and growth rates. We have used the concepts and methods of linear systems, algebra, and latent semantic analysis to solve this problem step by step. We have also learned some tips and tricks to make the problem easier and more fun. Here are the main points that we have covered:

• Convert the feet to inches for both types of trees
• Set up an equation for each type of tree
• Solve for the number of years when the trees are the same height
• Justify the answer by checking the original equations

We hope that this article has helped you understand and apply the concepts and methods of linear systems, algebra, and latent semantic analysis to solve similar word problems.

Thank you for reading this article and happy gardening!