Lesson 3.8: Solving More Equations

QuantwayTM Student Handout

Lesson 3.8: Solving More Equations

Theme: Personal Finance, Medical Literacy, Citizenship

Specific Objectives

Students will understand that

· solving all equations follows the basic rules of undoing and keeping the equation balanced.

Students will be able to

· solve linear equations that require simplification before solving.

· solve for a variable in a linear equation in terms of another variable.

· solve for a variable in a single-term quadratic equation.

Problem Situation: Solving Equations of Different Forms

Solving equations such as the Widmark equation for blood alcohol content (BAC) and proportional equations for resizing graphics is an important skill. Mathematical models are often constructed to represent real-life situations. Being able to use these equations fully includes being able to solve for various unknown variables in the equation. Below are three scenarios for you to practice and enhance your equation-solving skills. With each answer, check that the answer is reasonable given the context and that you have included the correct units with your solution.

(1) Paula has two options for going to school. She can carpool with a friend or take the bus. Her friend estimates that driving will cost 22 cents per mile for gas and 8.2 cents per mile for maintenance of the car. Additionally, there is a $25 parking fee per week at the college. If Paula carpools, she would pay half of these costs. The cost of the carpool can be modeled by the following equation where C is cost of carpooling per week and m is the total miles driven to school each week:

(a) Explain what each term in the equation represents.

(b) Find the total weekly carpooling cost if the commute to school is 7 miles each way and Paula goes to school three times a week.

(c) A weekly bus pass costs $22.00 dollars. How many total miles must Paula commute to school each week for the carpool cost to be equal to the bus pass? How many trips to school each week must Paula make for the bus pass to be less expensive than carpooling?

(2) Recall Widmark’s equation for BAC. In the case of the average male who weighs 190 pounds,[1] you can simplify Widmark’s formula to get

B = −0.015t + 0.022N

Forensic scientists often use this equation at the time of an accident to determine how many drinks someone had. In these cases, time (t) and BAC (B) are known from the police report. The crime lab uses this equation to estimate the number of drinks (N).

(a) Find the number of drinks if the BAC is 0.09 and the time is 2 hours.

(b) Since they use the formula to solve for N over and over, it is easier if the formula is rewritten so that it is solved for N. In other words, so that N is isolated on one side of the equation and all other terms are on the other side. Solve for N in terms of t and B.

(3) You volunteer for a nonprofit organization interested in women’s issues. The logo for your nonprofit organization is three identical squares arranged as follows:

(a) The organization wants to make banners of different sizes. Find an equation that can be used to find the total area of the logo based on the length of the side of one of the squares.

(b) The organization is sponsoring a walk-a-thon to raise funds for breast cancer research. You
want to recreate this logo in the middle of the racetrack with bras that have been collected at multiple drop-off sites around the city. You estimate that approximately 1,500 square feet of bras have been donated. How long should you make each side of the square?

Making Connections

Record the important mathematical ideas from the discussion.

Further Applications

(1) An artist is creating a sculpture using a sphere made of clay to represent Earth. The volume of a sphere is given by the equation:

where r is the radius of the sphere. The artist has a rectangular slab of clay that is 4 inches wide, 6 inches long, and 2 inches high. What is the radius of the largest sphere the artist can create with this clay?

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This lesson is part of QUANTWAY™, A Pathway Through College-Level Quantitative Reasoning, which is a product of a Carnegie Networked Improvement Community that seeks to advance student success. The original version of this work, version 1.0, was created by The Charles A. Dana Center at The University of Texas at Austin under sponsorship of the Carnegie Foundation for the Advancement of Teaching. This version and all subsequent versions, result from the continuous improvement efforts of the Carnegie Networked Improvement Community. The network brings together community college faculty and staff, designers, researchers and developers. It is a research and development community that seeks to harvest the wisdom of its diverse participants through systematic and disciplined inquiry to improve developmental mathematics instruction. For more information on the QuantwayTM Networked Improvement Community, please visit carnegiefoundation.org.

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Version 1.5

[1]Retrieved from www.cdc.gov