(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?

Introduction

Mathematics plays a crucial role in our daily lives, often in ways that may not be immediately apparent. This essay explores various mathematical problems and applications in real-life scenarios. These scenarios include linear modeling of tree growth, profit calculation, coordinate geometry, parallel lines, perpendicular lines, and financial planning for mobile phone plans. Each scenario highlights the practical relevance of mathematical concepts in solving everyday problems.

1. Linear Modeling of Tree Growth

In 1990, as part of Earth Awareness Week, six 2-foot tall pine trees were planted. These trees have been growing at an average rate of 34 feet per year. To understand the height of these trees in relation to the number of years since they were planted, we can create a linear model. A linear model is a mathematical representation that relates two variables with a linear equation of the form y = mx + b.

In this case, let’s use the following variables:

• y represents the height of the trees in feet.
• x represents the number of years since the trees were planted.

To find the linear model, we need to determine the values of the slope (m) and the y-intercept (b) in the equation y = mx + b.

The initial height of the trees in 1990 was 2 feet, so we have a data point: (0, 2). This data point allows us to find the y-intercept (b) in the linear model. Using the point-slope form of a linear equation:

y−y1=m(x−x1)y−y1=m(x−x1),

where (x₁, y₁) is a known point on the line (0, 2 in this case), we can substitute the values:

$y-2=m(x-0)$.

Simplifying:

$y-2=mx$.

Now, we can find the slope (m) using the average growth rate of 34 feet per year. The slope represents the change in height (y) for every year (x). So:

m=Change in heightChange in years=34 ft1 year=34m=Change in yearsChange in height=1 year34 ft=34.

Now, we have both the slope and the y-intercept. The linear model for the height of the trees is:

$y=34x+2$.

In this equation, y represents the height of the trees, and x represents the number of years since the trees were planted.

2. Profit Calculation

Erin is preparing for her family reunion and plans to make and sell 30 shirts. She intends to sell each shirt for $18 apiece, and the cost to make each shirt is $10. To calculate her profit, we can use the following formula:

$Profit=(SellingPriceperShirt-CostperShirt)\times NumberofShirtsSold$.

In this case, Erin sells 25 shirts, so the profit can be calculated as:

Profit = (18 - 10) imes 25 = 8 imes 25 = $200.

Erin makes a profit of $200 from selling 25 shirts at the family reunion.

3. Coordinate Geometry: Midpoint Calculation

Billy’s hometown is mapped on a coordinate grid with the origin at City Hall. Billy’s house is located at the point (8, 7), and his best friend’s house is located at (2, -1). To find the midpoint between the two houses, we can use the midpoint formula:

(xmidpoint, ymidpoint)=(x1+x22, y1+y22)(xmidpoint, ymidpoint)=(2x1+x2, 2y1+y2).

Substituting the coordinates of Billy’s and his best friend’s houses:

(xmidpoint, ymidpoint)=(8+22, 7+(−1)2)=(102, 62)=(5,3)(xmidpoint, ymidpoint)=(28+2, 27+(−1))=(210, 26)=(5,3).

The midpoint between the two houses is located at (5, 3).

4. Coordinate Geometry: Distance Calculation

In this scenario, we have a coordinate grid representing a math class. Mary’s seat is located at (-4, 7), and Betty’s seat is located at (-2, 5). To calculate the distance between Mary’s and Betty’s seats, we can use the distance formula:

Distance=(x2−x1)2+(y2−y1)2Distance=(x2−x1)2+(y2−y1)2

.

Substituting the coordinates of Mary’s and Betty’s seats:

Distance=(−2−(−4))2+(5−7)2=(2)2+(−2)2=4+4=8.Distance=(−2−(−4))2+(5−7)2

=(2)2+(−2)2=4+4=8

.

The distance between Mary’s and Betty’s seats is 88

 units.

5. Parallel Lines and Slope-Intercept Form

In this problem, we have the equation of 1st Street as $y=6x-7$, and we are tasked with finding the equation for 2nd Street, which is parallel to 1st Street and passes through the point (2, 9).

Parallel lines have the same slope. Therefore, the slope of 2nd Street will also be 6, as it is parallel to 1st Street. To find the equation of 2nd Street in slope-intercept form (y = mx + b), we need to find the y-intercept (b). We can use the point (2, 9) on 2nd Street to find b:

$9=6(2)+b$.

Solving for b:

$b=9-12=-3$.

Therefore, the equation for 2nd Street in slope-intercept form is:

$y=6x-3$.

6. Perpendicular Lines

As a construction manager, you are tasked with building a new road that crosses the point (1, 1). There is already an existing road with the equation $y=3x-1$. You are asked to build your road such that it crosses the existing road at a perpendicular angle. To find the equation of your road, we need to determine its slope and then use the point-slope form.

The existing road has a slope of 3. To find a perpendicular line, we can use the negative reciprocal of the slope, which is $m=-1/3$. Now, we have the slope for your road.

Next, we can use the point-slope form of a line:

y−y1=m(x−x1)y−y1=m(x−x1),

where (x₁, y₁) is the point (1, 1) in this case, and m is the slope (-1/3):

y−1=−13(x−1)y−1=−31(x−1).

Now, let’s simplify the equation:

y−1=−13x+13y−1=−31x+31.

To isolate y, we add 1 to both sides:

y=−13x+43y=−31x+34.

Therefore, the equation of your road is:

y=−13x+43y=−31x+34.

This equation represents your road, which crosses the existing road at a perpendicular angle.

7. Financial Planning for Mobile Phone Plans

In this scenario, we have two mobile phone plans: the Family Plan and the Mobile Share Plan. Let’s analyze these plans and answer the given questions:

a. The Family Plan:

• Monthly fee: $90
• Data charges: $40 per device for up to 2 GB of data per device.
• Unlimited talk and text on up to 8 lines.

To find the model of the total cost of the Family Plan (C), we can use the following formula, where P is the number of devices that need data plans:

$C=Monthlyfee+Datachargesperdevice\times P$.

Substituting the given values:

$C=90+40P$.

b. The Mobile Share Plan:

• Monthly fee: $120
• Data charges: $35 per device up to a shared total of 10 GB of data.
• Unlimited talk and text for all the lines.

To find the model of the total cost of the Mobile Share Plan (C), we need to consider the data charges based on the number of devices and the shared data limit:

• If P devices need data plans, and the total shared data is 10 GB, the data charges will be $35\times P$ if $P\le 10$ and $35\times 10$ if $P>10$.

Now, we can construct the model for the total cost of the Mobile Share Plan (C):

C=120+{35P,if P≤1035×10,if P>10C=120+{35P,35×10,if P≤10if P>10.

c. To find the number of devices that would make the two plans equal in cost, we need to set the two cost models equal to each other and solve for P:

90+40P=120+{35P,if P≤1035×10,if P>1090+40P=120+{35P,35×10,if P≤10if P>10.

Let’s consider two cases:

Case 1: $P\le 10$

In this case, we have:

$90+40P=120+35P$.

Subtract 35P from both sides:

$90+5P=120$.

Subtract 90 from both sides:

$5P=30$.

Divide by 5:

$P=6$.

So, if the number of devices (P) is 6 or less, the Family Plan is cheaper.

Case 2: $P>10$

In this case, we have:

$90+40P=120+35\times 10$.

Simplify:

$90+40P=120+350$.

Subtract 120 from both sides:

$40P=350-90$.

$40P=260$.

Divide by 40:

$P=6.5$.

Since the number of devices cannot be a fraction, we disregard this case.

Therefore, the two plans will be equal in cost if the number of devices (P) is 6.

d. If a family has 3 smartphones, which plan should they choose?

To determine which plan is more cost-effective for a family with 3 smartphones, we can calculate the cost for each plan using P = 3 (3 devices).

For the Family Plan:

C_{ ext{Family Plan}} = 90 + 40 imes 3 = 90 + 120 = $210.

For the Mobile Share Plan:

C_{ ext{Mobile Share Plan}} = 120 + 35 imes 3 = 120 + 105 = $225.

In this case, the Family Plan would be the more cost-effective choice for the family with 3 smartphones, as it costs $210 compared to the Mobile Share Plan’s $225.

Conclusion

In this essay, we explored various real-life scenarios that required mathematical modeling and problem-solving. We applied mathematical concepts such as linear equations, coordinate geometry, slope-intercept form, and cost modeling to address practical problems. These examples illustrate how mathematics plays an essential role in everyday life, from predicting tree growth to making financial decisions about mobile phone plans. By applying mathematical principles, we can better understand and solve a wide range of real-world challenges.