Equations and Inequalities

Chapter 2

Equations and Inequalities

2.3 Models and Applications

1. Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?

[pic 1]

Mark: [pic 2], Don: [pic 3]

1. Beth and Ann are joking that their combined ages equal Sam’s age. If Beth is twice Ann’s age and Sam is 69 yr old, what are Beth and Ann’s ages?
   [pic 4]
   Ann: [pic 5] Beth: [pic 6]
   

1. Ben originally filled out 8 more applications than Henry. Then each boy filled out 3 additional applications, bringing the total to 28. How many applications did each boy originally fill out?
   [pic 7]
   7 and 15

For the following exercises, use this scenario: Two different telephone carriers offer the following plans that a person is considering. Company A has a monthly fee of $20 and charges of $0.05/min for calls. Company B has a monthly fee of $5 and charges $0.10/min for calls.

1. Find the model of the total cost of Company A’s plan, using [pic 8] for the minutes.
   [pic 9] 
2. Find the model of the total cost of Company B’s plan, using [pic 10] for the minutes.
   [pic 11] 
3. Find out how many minutes of calling would make the two plans equal.
   [pic 12][pic 13]
   300 min
4. If the person makes a monthly average of 200 min of calls, which plan should for the person choose?
   [pic 14]
   Company B

For the following exercises, use this scenario: A wireless carrier offers the following plans that a person is considering. The Family Plan: $90 monthly fee, unlimited talk and text on up to 5 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 [pic 15] for the number of devices that need data plans as part of their cost.

1. Find the model of the total cost of the Family Plan.
   [pic 16] 
2. Find the model of the total cost of the Mobile Share Plan.
   [pic 17] 
3. Assuming they stay under their data limit, find the number of devices that would make the two plans equal in cost.
   [pic 18][pic 19]
   6 devices
4. If a family has 3 smart phones, which plan should they choose?
   [pic 20]
   The Family Plan

1. Two planes fly in opposite directions. One travels 450 mi/h and the other 550 mi/h. How long will it take before they are 4,000 mi apart?
   [pic 21]
   4 h

1. Ben starts walking along a path at 4 mi/h. One and a half hours after Ben leaves, his sister Amanda begins jogging along the same path at 6 mi/h. How long will it be before Amanda catches up to Ben?
   [pic 22][pic 23]
   3 h
2. Fiora starts riding her bike at 20 mi/h. After a while, she slows down to 12 mi/h, and maintains that speed for the rest of the trip. The whole trip of 70 mi takes her 4.5 h. For what distance did she travel at 20 mi/h?
   [pic 24]
   She traveled for 2 h at 20 mi/h, or 40 miles.
3. A chemistry teacher needs to mix a 30% salt solution with a 70% salt solution to make 20 qt of a 40% salt solution. How many quarts of each solution should the teacher mix to get the desired result?
   [pic 25]
   15 qt of 30% solution and 5 qt of the 70% solution

For the following exercises, use this scenario: A truck rental agency offers two kinds of plans. Plan A charges $75/wk plus $.10/mi driven. Plan B charges $100/wk plus $.05/mi driven.

1. Write the model equation for the cost of renting a truck with plan A.
   [pic 26] 
2. Write the model equation for the cost of renting a truck with plan B.
   [pic 27]
3. Find the number of miles that would generate the same cost for both plans.
   [pic 28][pic 29]
   500 mi
4. If Tim knows he has to travel 300 mi, which plan should he choose?
   [pic 30]
   Plan A

For the following exercises, solve for the given variable in the formula. After obtaining a new version of the formula, you will use it to solve a question.

1. Solve for W: [pic 31] 
   
   [pic 32][pic 33]
   [pic 34] 
2. Use the formula from the previous question to find the width, [pic 35] of a rectangle whose length is 15 and whose perimeter is 58.
   [pic 36]
3. Solve for [pic 37] [pic 38] 
   [pic 39]
   [pic 40] 
4. Use the formula from the previous question to find [pic 41] when [pic 42]
   [pic 43]