Javier's Shrub Was 75 Centimeters Tall When He Planted It And Has Grown 68 Centimeters Per Year Since.Let $y$ Represent The Number Of Years Since Javier Planted The Shrub And $h$ Represent The Height Of The Shrub In
Introduction
In this article, we will delve into the world of mathematics to understand the growth of Javier's shrub. We will use algebraic equations to model the height of the shrub over time, taking into account its initial height and annual growth rate. By the end of this discussion, you will have a clear understanding of how to represent the growth of the shrub mathematically.
Initial Conditions
Let's start by understanding the initial conditions of the problem. Javier's shrub was 75 centimeters tall when he planted it. This is our starting point, and we will use this information to develop an equation that represents the height of the shrub over time.
Annual Growth Rate
The shrub has grown 68 centimeters per year since it was planted. This means that every year, the height of the shrub increases by 68 centimeters. We can use this information to develop an equation that represents the height of the shrub at any given time.
Mathematical Modeling
To model the growth of the shrub, we can use a linear equation. A linear equation is a mathematical equation in which the highest power of the variable(s) is 1. In this case, we have one variable, y, which represents the number of years since Javier planted the shrub. We can use the following equation to represent the height of the shrub:
h(y) = 75 + 68y
In this equation, h(y) represents the height of the shrub at time y, and 75 is the initial height of the shrub. The term 68y represents the annual growth rate of the shrub.
Interpreting the Equation
Let's break down the equation and understand what it represents. The equation h(y) = 75 + 68y tells us that the height of the shrub at time y is equal to the initial height (75) plus the product of the annual growth rate (68) and the number of years (y). This means that every year, the height of the shrub increases by 68 centimeters.
Example Calculations
Let's use the equation to calculate the height of the shrub at different times. For example, if we want to know the height of the shrub after 5 years, we can plug in y = 5 into the equation:
h(5) = 75 + 68(5) h(5) = 75 + 340 h(5) = 415
This tells us that after 5 years, the height of the shrub will be 415 centimeters.
Graphing the Equation
We can also graph the equation to visualize the growth of the shrub over time. The graph will be a straight line, with the initial height (75) as the starting point and the annual growth rate (68) as the slope.
Conclusion
In this article, we used algebraic equations to model the growth of Javier's shrub. We developed a linear equation that represents the height of the shrub at any given time, taking into account its initial height and annual growth rate. By understanding the mathematical model, we can predict the height of the shrub at different times and visualize its growth over time.
Growth of the Shrub Over Time
Height of the Shrub Over Time
| Year | Height (cm) |
|---|---|
| 0 | 75 |
| 1 | 143 |
| 2 | 211 |
| 3 | 279 |
| 4 | 347 |
| 5 | 415 |
| 6 | 483 |
| 7 | 551 |
| 8 | 619 |
| 9 | 687 |
Graph of the Shrub's Growth
Mathematical Representation of the Shrub's Growth
The growth of the shrub can be represented mathematically using the following equation:
h(y) = 75 + 68y
Where h(y) is the height of the shrub at time y, and 75 is the initial height of the shrub. The term 68y represents the annual growth rate of the shrub.
Understanding the Mathematical Model
The mathematical model represents the growth of the shrub as a linear equation. This means that the height of the shrub increases by a constant amount (68 centimeters) every year. The model can be used to predict the height of the shrub at different times and visualize its growth over time.
Real-World Applications
The mathematical model of the shrub's growth can be applied to real-world situations, such as:
- Predicting the growth of plants in a garden or greenhouse
- Modeling the growth of animals in a population
- Understanding the growth of cities or populations over time
Conclusion
Q: What is the initial height of Javier's shrub?
A: The initial height of Javier's shrub is 75 centimeters.
Q: How much does the shrub grow per year?
A: The shrub grows 68 centimeters per year.
Q: What is the equation that represents the height of the shrub over time?
A: The equation that represents the height of the shrub over time is:
h(y) = 75 + 68y
Where h(y) is the height of the shrub at time y, and 75 is the initial height of the shrub.
Q: How can I use the equation to predict the height of the shrub at a specific time?
A: To predict the height of the shrub at a specific time, simply plug in the value of y (the number of years since the shrub was planted) into the equation. For example, if you want to know the height of the shrub after 5 years, you would plug in y = 5:
h(5) = 75 + 68(5) h(5) = 75 + 340 h(5) = 415
This tells us that after 5 years, the height of the shrub will be 415 centimeters.
Q: Can I use the equation to predict the height of the shrub at any time, not just after a whole number of years?
A: Yes, you can use the equation to predict the height of the shrub at any time, not just after a whole number of years. For example, if you want to know the height of the shrub after 3.5 years, you would plug in y = 3.5:
h(3.5) = 75 + 68(3.5) h(3.5) = 75 + 238 h(3.5) = 313
This tells us that after 3.5 years, the height of the shrub will be 313 centimeters.
Q: Can I use the equation to compare the growth of different shrubs?
A: Yes, you can use the equation to compare the growth of different shrubs. For example, if you have two shrubs with different initial heights and growth rates, you can use the equation to predict their heights at different times.
Q: What are some real-world applications of the equation?
A: The equation can be used in a variety of real-world applications, such as:
- Predicting the growth of plants in a garden or greenhouse
- Modeling the growth of animals in a population
- Understanding the growth of cities or populations over time
Q: Can I use the equation to make predictions about the future growth of the shrub?
A: Yes, you can use the equation to make predictions about the future growth of the shrub. For example, if you want to know the height of the shrub after 10 years, you can plug in y = 10:
h(10) = 75 + 68(10) h(10) = 75 + 680 h(10) = 755
This tells us that after 10 years, the height of the shrub will be 755 centimeters.
Conclusion
In this article, we answered some frequently asked questions about the growth of Javier's shrub. We discussed the initial height of the shrub, its annual growth rate, and the equation that represents the height of the shrub over time. We also explored some real-world applications of the equation and how it can be used to make predictions about the future growth of the shrub.