Assignment Question
Answers must be accompanied by all work in order to receive credit. Answers by themselves will receive ZERO (0) credit. ** (1) Six 2 foot tall pine trees were planted during the school’s observation of Earth Awareness Week in 1990. The trees have grown at an average rate of 34 foot per year. Write a linear model that gives the height of the trees in terms of the number of years since they were planted, and tell what the y variable represent in the equation ? (2) Erin is making thirty shirts for her upcoming family reunion. At the reunion she is selling each shirt for $18 apiece. If each shirt cost her $10 apiece to make, how much profit does she make if she only sells 25 shirts at the reunion? (3) Billy’s hometown is mapped on a coordinate grid with the origin being at City Hall. Billy’s house is located at the point (8, 7) and his best friend’s house is located at (2, -1). Where is the midpoint between the two houses located? (4) The class of math is mapped on a coordinate grid with the origin being at the center point of the hall. Mary’s seat is located at the point (-4, 7) and Betty’s seat is located at (-2, 5). How far is it from Mary’s seat to Betty’s seat? (5) When put on a coordinate plane, 1st street has the equation y = 6x – 7. 2nd street is parallel to 1st street and goes through the point (2, 9). Write the equation for 2nd Street in slope-intercept form. (6) As a construction manager, you are asked to build a new road, which crosses the point (1,1). There is another road already built, which can be expressed as y=3x−1. You are asked to build your road such that it crosses this road at a perpendicular angle. Find the equation of your road. Leave answer as a fraction or mixed number if needed. (7) The Family Plan: $90 monthly fee, unlimited talk and text on up to 8 lines, and data charges of $40 for each device for up to 2 GB of data per device. The Mobile Share Plan: $120 monthly fee for up to 10 devices, unlimited talk and text for all the lines, and data charges of $35 for each device up to a shared total of 10 GB of data. Use P for the number of devices that need data plans as part of their cost. a. Find the model of the total cost of the Family Plan. b. Find the model of the total cost of the Mobile Share Plan. c. Assuming they stay under their data limit, find the number of devices that would make the two plans equal in cost. d. If a family has 3 smart phones, which plan should they choose?
Assignment Answer
In this mathematical problem-solving paper, we will address a variety of math questions covering linear models, midpoints, and cost comparisons. Each problem will be analyzed step by step to find the solution. Let’s dive into these math challenges:
Problem 1
Six 2-foot tall pine trees were planted during the school’s observation of Earth Awareness Week in 1990. The trees have grown at an average rate of 3.4 feet per year.
Linear Model: The height of the trees (y) in terms of the number of years since they were planted (x) can be represented by the equation: y = 2 + 3.4x
Here, the variable ‘y’ represents the height of the trees, and ‘x’ represents the number of years since they were planted.
Problem 2
Erin is making thirty shirts for her upcoming family reunion. She sells each shirt for $18 apiece, and each shirt costs her $10 apiece to make. If she only sells 25 shirts at the reunion, how much profit does she make?
To find the profit, we calculate the total revenue and subtract the total cost: Total Revenue = 25 shirts * $18 = $450 Total Cost = 25 shirts * $10 = $250 Profit = Total Revenue – Total Cost = $450 – $250 = $200
Problem 3
Billy’s house is at the point (8, 7), and his best friend’s house is at (2, -1). Where is the midpoint between the two houses located?
Midpoint Formula: The midpoint (x, y) between two points (x1, y1) and (x2, y2) is given by: x = (x1 + x2) / 2 y = (y1 + y2) / 2
Using the coordinates, we find: x = (8 + 2) / 2 = 10 / 2 = 5 y = (7 + (-1)) / 2 = 6 / 2 = 3
So, the midpoint is located at (5, 3).
Problem 4
Mary’s seat is at the point (-4, 7), and Betty’s seat is at (-2, 5). How far is it from Mary’s seat to Betty’s seat?
Distance Formula: The distance (d) between two points (x1, y1) and (x2, y2) is given by: d = √((x2 – x1)^2 + (y2 – y1)^2)
Using the coordinates, we calculate: d = √((-2 – (-4))^2 + (5 – 7)^2) = √(2^2 + (-2)^2) = √(4 + 4) = √8
So, the distance between Mary’s seat and Betty’s seat is √8 units.
Problem 5
When put on a coordinate plane, 1st street has the equation y = 6x – 7. 2nd street is parallel to 1st street and goes through the point (2, 9). Write the equation for 2nd Street in slope-intercept form.
Since 2nd street is parallel to 1st street, it has the same slope, which is 6. Using the point-slope form, we can write the equation for 2nd Street: y – 9 = 6(x – 2)
Now, let’s simplify it to slope-intercept form: y – 9 = 6x – 12 y = 6x – 12 + 9 y = 6x – 3
So, the equation for 2nd Street is y = 6x – 3.
Problem 6
As a construction manager, you are asked to build a new road that crosses the point (1, 1). There is another road already built, which can be expressed as y = 3x – 1. You are asked to build your road such that it crosses this road at a perpendicular angle. Find the equation of your road.
To make the roads perpendicular, the slopes must be negative reciprocals of each other. The slope of the existing road is 3. The negative reciprocal is -1/3.
Using the point-slope form, we can write the equation for the new road: y – 1 = (-1/3)(x – 1)
Now, simplify it to slope-intercept form: y – 1 = (-1/3)x + 1/3
Add 1 to both sides: y = (-1/3)x + 1/3 + 1 y = (-1/3)x + 4/3
So, the equation of your road is y = (-1/3)x + 4/3.
Problem 7
The Family Plan costs $90 monthly fee, plus data charges of $40 for each device for up to 2 GB of data per device. The Mobile Share Plan costs $120 monthly fee for up to 10 devices, plus data charges of $35 for each device up to a shared total of 10 GB of data.
a. Model of the total cost of the Family Plan: Total Cost = $90 (monthly fee) + $40P (data charges for each device)
b. Model of the total cost of the Mobile Share Plan: Total Cost = $120 (monthly fee) + $35P (data charges for each device, up to 10 devices)
c. To find the number of devices that would make the two plans equal in cost, set the total costs equal to each other: $90 + $40P = $120 + $35P
Now, solve for P (number of devices): $40P – $35P = $120 – $90 $5P = $30 P = $30 / $5 P = 6
So, if they have 6 devices, both plans will cost the same.
d. If a family has 3 smartphones, which plan should they choose? For 3 devices, let’s compare the costs: Family Plan: Total Cost = $90 + $40 * 3 = $90 + $120 = $210 Mobile Share Plan: Total Cost = $120 + $35 * 3 = $120 + $105 = $225
They should choose the Family Plan, as it is cheaper for 3 devices.
Frequently Asked Questions (FAQs)
FAQ 1: What is the linear model for the height of pine trees planted in 1990, and what does the ‘y’ variable represent in the equation?
FAQ 2: How much profit does Erin make if she sells 25 shirts at her family reunion, considering the cost of making each shirt?
FAQ 3: Where is the midpoint located between Billy’s house at (8, 7) and his best friend’s house at (2, -1)?
FAQ 4: What is the distance between Mary’s seat at (-4, 7) and Betty’s seat at (-2, 5)?
FAQ 5: How do you find the equation for the second street, which is parallel to the first street and passes through the point (2, 9)?
