Meadows or Malls Cover Letter
By: Grady James
We began this unit by learning how to solve systems of equations with 2 variables using several methods. First, we started by learning the elimination method of solving systems of equations. Then we learned the substitution method of solving systems of 2 equations and 2 variables. To solve you also could graph the functions and find where they intersected to solve for the system of equations. After this, we were introduced to systems of equations involving 3 variables. First, we began learning about these systems by visualizing them by making physical 3D graphs out of cardboard with X, Y, Z axis. This visualization using a physical model would help the conceptualization of the complex systems of equations with 3 equations and variables. Through this, we learned that it is necessary to have 3 variables in order to solve for a 3-dimensional graph. This is because one of these graphs takes the form of a plane, and when you have only two functions intersecting the two planes intersect along a line. A line is not a definable value thus you need 3 planes intersecting to solve for a singular point among the system of equations. To solve for these more complex systems was difficult to impossible using the previous methods used to solve the 2 variable systems of equations. This lead way into the study of matrices and their implications for solving systems of equations. To understand matrices the class began by first figuring out how matrices are organized, and how that organization relates to systems of equations. Here is an example of one system of equations the class might have practiced turning into a matrix. 2X-Y=12 [2 -1:12] Using this method of organization several
4X-7Y=48 [4 -7:48] functions could be performed on the values of the system of equations. The class learned that when multiplying matrices they needed to contain certain dimensions to multiply. The inside dimensions when written like this 2x1
1x4 need to match in order to multiply. We also learned how to add, subtract, divide and take the inverse of matrices in order to solve for certain aspects of systems of equations. Using matrices, several situations and word problems containing systems of equations and constraints similar to that of the unit problem could now be solved. Matrices gave us the necessary tool to solve for the complex unit problem. During this unit, I enjoyed the refresher on matrices and complex systems of equations. Matrices can be a valuable skill when balancing multiple functions. Matrices can be a valuable skill that can be applied to real-world applications. One way that I will be needing to solve for a system of equations is when I am programming an Arduino microcontroller to solve the system of equations. This system is where the Arduino will be calculating its acceleration towards the ground under one parachute and then be solving for the common point where the current acceleration towards the ground and the acceleration after it deploys the second parachute are true to the landing time.
4X-7Y=48 [4 -7:48] functions could be performed on the values of the system of equations. The class learned that when multiplying matrices they needed to contain certain dimensions to multiply. The inside dimensions when written like this 2x1
1x4 need to match in order to multiply. We also learned how to add, subtract, divide and take the inverse of matrices in order to solve for certain aspects of systems of equations. Using matrices, several situations and word problems containing systems of equations and constraints similar to that of the unit problem could now be solved. Matrices gave us the necessary tool to solve for the complex unit problem. During this unit, I enjoyed the refresher on matrices and complex systems of equations. Matrices can be a valuable skill when balancing multiple functions. Matrices can be a valuable skill that can be applied to real-world applications. One way that I will be needing to solve for a system of equations is when I am programming an Arduino microcontroller to solve the system of equations. This system is where the Arduino will be calculating its acceleration towards the ground under one parachute and then be solving for the common point where the current acceleration towards the ground and the acceleration after it deploys the second parachute are true to the landing time.
Orchard Hideout
By: Grady James
The central problem of this unit is centered around a orchard planted in rows in a circle with a radius of 50 units. On each intersection, there is a tree there is a point at the center which is supposed to be a hideout at a certain diameter of the trees so it blocks all lines of sight out of the orchard. The circumference of circles is related to this problem in the sense that we are looking at the tree's growth as it the increases the radius of the trunk. We are dealing with a circular orchard so circles are related to the problem in that way as well. The distance between points was used to solve for the radius of trees, and where they met the slope of a line that has the last line of sight out of the orchard. The last sight line is the line which is the last linear line from the center of the orchard which is not blocked by a tree trunk. The orchard problem can be solved using some right triangles and trigonometry fairly easy to figure out the slope of the last sight line. The radius at the last line of sight is blocked can be solved for in a very similar manner as seen in my work for solving for the unit problem included in the documents in this mathematical portfolio.
During the course of this unit, I gained a greater understanding of how to combine multiple geometric equations and use algebra to solve for desired geometric values in a problem. An example of this is when in class we derived the midpoint formula from the Pythagorean Theorem a geometric equation. This unit has been a necessary and an informative venture on topics that were rushed in their explanation in my courses previously.
During the course of this unit, I gained a greater understanding of how to combine multiple geometric equations and use algebra to solve for desired geometric values in a problem. An example of this is when in class we derived the midpoint formula from the Pythagorean Theorem a geometric equation. This unit has been a necessary and an informative venture on topics that were rushed in their explanation in my courses previously.