Problem 48 - Algebra 1

Question

You plant a 14 -inch hemlock tree in your backyard that grows at a rate of 4 inches per year and an 8 -inch blue spruce tree that grows at a rate of 6 inches per year. In how many years after you plant the trees will the two trees be the same height? How tall will each tree be?

Step-by-Step Solution

Answer

The two trees will be the same height 3 years from now. Both trees will be 26 inches tall.

Step 1: Formulate Equations

Start by formulating equations that represent how the heights of the trees change every year. Say 'x' is the number of years from now. The height of hemlock tree in 'x' years can be represented by the equation \(H = 14 + 4x\), while the height of the blue spruce tree can be represented by the equation \(B = 8 + 6x\).

Step 2: Solve the Equation

The moment when the heights of the trees are equal is when \(H = B\). Replacing 'H' and 'B' with the aforementioned formulas, this provides: \(14 + 4x = 8 + 6x\). Solve the equation by isolating 'x'.

Step 3: Find 'x'

By simplifying the equation, you find that \(2x = 6\), So, \(x = 3\). This means that the trees will be the same height 3 years from now.

Step 4: What is the Height?

To find the height, substitute 'x' into either of the original equations (it should yield the same height for both trees). Using the hemlock tree equation: \(H = 14 + 4 * 3 = 26\), the height of both trees in 3 years will be 26 inches.

Key Concepts

Algebraic Expressions

Algebraic expressions are combinations of variables, numbers, and arithmetic operations like addition, subtraction, multiplication, and division. These expressions are fundamental building blocks when dealing with equations and modeling real-world scenarios in algebra. For example, the height of a growing tree over time can be represented as an algebraic expression.

In the exercise, the height of a hemlock tree after a certain number of years is represented by the algebraic expression \( H = 14 + 4x \), where \( H \) is the height of the tree in inches and \( x \) is the number of years passed. Similarly, the height of a blue spruce tree is given by \( B = 8 + 6x \). These expressions enable us to create models of linear growth, which we can manipulate to solve for specific conditions, such as when both trees reach the same height.

Equations of Linear Growth

Equations of linear growth describe situations where the rate of change is constant. In the provided exercise, the growth of both trees is consistent annually, making this a perfect example of linear growth.

For the hemlock tree, the linear equation is \( H = 14 + 4x \), and for the blue spruce, it's \( B = 8 + 6x \). The coefficients of \( x \) – 4 and 6 – represent the growth rate in inches per year for the hemlock and blue spruce trees, respectively. To determine when the trees will be the same height, we need to find the value of \( x \) at which the expressions for \( H \) and \( B \) are equal. This is a straightforward example of how linear equations are used to model and solve real-world problems.

Systems of Equations

Systems of equations involve two or more equations with the same variables. The goal is to find a solution that satisfies all equations simultaneously. In our tree growth problem, we're working with a simple system of two equations: \( H = 14 + 4x \) and \( B = 8 + 6x \).

To find when the trees are the same height, we set these two equations equal to each other to determine the value of \( x \) that makes both heights equal. The system is solved by substituting one equation into the other, or by equating the expressions and solving for \( x \), which leads us to the number of years until the trees are the same height. Solving systems of equations is an essential skill in algebra, with applications in science, economics, engineering, and more.