Math 6
Mathematics homework help
Lesson 3.2
Introduction
Course Objectives
This lesson will address the following course outcomes:
· 1. Demonstrate operation sense by communicating in words and symbols the effects of operations on numbers. Apply the correct order of operations in evaluating expressions and formulas.
· 13. Evaluate formulas with multiple variables in a variety of contexts, including science, statistics, geometry, and financial math. Solve simple formulas for a specified variable.
· 15. Solve linear equations in one variable, including problems involving the distributive property and fractions.
Specific Objectives
Students will understand that
· the behavior of a formula can be explored using a table and graph.
Students will be able to
· simplify a formula given values for some parameters.
· solve for a variable in a linear equation, and evaluate an equation.
Problem Situation: Calculating Blood Alcohol Content
Blood alcohol content (BAC) is a measurement of how much alcohol is in someone’s blood. It is usually measured as a percentage. So, a BAC of 0.3% is three-tenths of 1%. That is, there are 3 grams of alcohol for every 1,000 grams of blood. A BAC of 0.05% impairs reasoning and the ability to concentrate. A BAC of 0.30% can lead to a blackout, shortness of breath, and loss of bladder control. In most states, the legal limit for driving is a BAC of 0.08%.
#1 Points possible: 12. Total attempts: 5
Think about the variables that might influence BAC, and how they would influence it. Use your intuition to answer these questions:
If you drink more drinks, your BAC would
If the time since your first drink was longer, your BAC would
If you weighed more, your BAC would
Estimating BAC
BAC is usually determined by a breathalyzer, urinalysis, or blood test. However, Swedish physician, E.M.P. Widmark developed the following equation for estimating an individual’s BAC. This formula is widely used by forensic scientists:
B=−0.015t+2.84NW⋅gB=-0.015t+2.84NW⋅g
Where
B = percentage of BAC
N = number of “standard drinks” (A standard drink is one 12-ounce beer, one 5-ounce glass of wine, or one 1.5-ounce shot of liquor.) N should be at least 1.
W = weight in pounds
g = gender constant, 0.68 for men and 0.55 for women
t = number of hours since the first drink
The variables B, N, W, g, and t change depending on the person and situation. The units of B are a percentage. So, 0.08 is not 8%, but eight-hundredths of a percent.
This equation has been simplified from the one found in the reference. The numbers 0.015 and 2.84 are constants based on the average person. It’s also worth noting many high-end beers have twice as much alcohol as the “standard beer,” which assumes 4% alcohol.
#2 Points possible: 5. Total attempts: 5
Consider the case of a male student who has three beers and weighs 120 pounds. For this case, these values can replace the appropriate variables in the formula. What variables are still unknown in the equation? (select all that are still unknown)
· B
· N
· W
· g
· t
#3 Points possible: 5. Total attempts: 5
Consider the case of a male student who has three beers and weighs 120 pounds. Simplify the Widmark equation as much as possible for this case, rounding any constants to 3 decimal places. Hint
#4 Points possible: 12. Total attempts: 5
Using your simplified equation, find the estimated BAC for this student one, three, and five hours after his first drink. Look for patterns in the data.
Hours | BAC | |
1 | % | |
3 | % | |
5 | % |
#5 Points possible: 6. Total attempts: 5
Complete the following contextual sentence:
Every hour, this student’s BAC by %.
#6 Points possible: 5. Total attempts: 5
Create a graph of the BAC over time for this student. Your graph doesn’t have to be perfect.
Clear All Draw:
#7 Points possible: 10. Total attempts: 5
Using your equation, what would this student’s BAC be after 8 hours?
%
Is this reasonable?
#8 Points possible: 10. Total attempts: 5
How long will it take for this student’s BAC to be 0.08%, the legal limit? Give your answer to 1 decimal place.
hours
How long will it take for the alcohol to be completely metabolized resulting in a BAC of 0.0? Give your answer to 1 decimal place.
hours
Hint: Try this problem on your own. If you’re having trouble after two tries, we’ll give some hints
Another Case
A female student, weighing 110 pounds, plans on going home in two hours. Using the Widmark formula, the simplified equation for this case is
B=−0.03+2.84N60.5B=-0.03+2.84N60.5
#9 Points possible: 8. Total attempts: 5
Compare her BAC for one glass of wine versus three glasses of wine at the time she will leave. Give BAC to 3 decimal places.
Number of Drinks | BAC | |
1 | % | |
3 | % |
#10 Points possible: 5. Total attempts: 5
In this scenario, determine how many drinks she can have so that her BAC remains less than 0.08%. Since she could finish only part of a drink, give your answer to one decimal place.
drinks
#11 Points possible: 5. Total attempts: 5
Create a graph of the BAC given number of drinks for this student. Your graph doesn’t have to be perfect.
Clear All Draw:
HW 3.2
#1 Points possible: 5. Total attempts: 5
Which of the following was one of the main mathematical ideas of the lesson?
· Blood Alcohol Content (BAC) is affected by many different variables.
· A formula can be explored by evaluating it and solving for specific values.
· When solving an equation, an operation that changes the value of one side must also be done to the other side of the equation.
· Sometimes letters can represent a constant.
#2 Points possible: 20. Total attempts: 5
Find the solution to each of the following:
a. 5+3x=145+3x=14 x =
b. 6x−5=106x-5=10 x =
c. 1=7+2×1=7+2x x =
d. x4+3=8×4+3=8 x =
#3 Points possible: 15. Total attempts: 5
Find the solution to each of the following:
a. 2.5x+1=8.52.5x+1=8.5 x =
b. 3.6+4.2x=9.93.6+4.2x=9.9 x =
c. 14.92=5+1.6×14.92=5+1.6x x =
#4 Points possible: 15. Total attempts: 5
Find the solution to each of the following, giving the answer in reduced form as proper or improper fractions (no mixed numbers).
a. 13x+38=5813x+38=58 x =
b. 12x+14=1612x+14=16 x =
c. 2x+23=82x+23=8 x =
#5 Points possible: 8. Total attempts: 5
Recall that Blood Alcohol Content (BAC) is a measurement of how much alcohol is in someone’s blood as a percentage. However, police and the public typically omit the language for % when quoting the BAC and simply say, “BAC is 0.04.”
Write an interpretation of what each of the following BAC values means in terms of how much alcohol is in the bloodstream in the form of the amount of alcohol per 1,000 grams of blood. You may want to refer back to the example in the lesson.
a. BAC = 0.1 gram of alcohol for every 1,000 grams of blood.
b. BAC = 0.02 gram of alcohol for every 1,000 grams of blood.
#6 Points possible: 1. Total attempts: 5
Use information from the website http://en.wikipedia.org/wiki/Blood_alcohol_content to list effects on an individual having a BAC as given. Give at least one effect for each.
a. BAC = 0.1
b. BAC = 0.5
c. BAC = 0.05
Use the Widmark Equation, B=−0.015t+2.84NW⋅gB=-0.015t+2.84NW⋅g to solve the next two questions. Recall that g = 0.68 for men and g = 0.55 for women.
#7 Points possible: 10. Total attempts: 5
A male student had five glasses of wine at a party. He weighs 155 pounds. Using the Widmark equation,
a) Write the equation you would solve to figure out how long it will take before his BAC is 0.08. Is it okay to round any constants to 3 decimal places.
b) Solve that equation. Round to 2 decimal places.
hours
#8 Points possible: 5. Total attempts: 5
This question will apply the Widmark Equation to you.
Enter your weight in pounds: pounds. Your gender:
Answer the next two questions based on the Widmark Equation using the weight and gender you specified.
If you drink alcohol over a period of 5 hours, how many whole drinks would you be able to consume and still ensure that your BAC is less than the legal limit in Washington? drinks
#9 Points possible: 10. Total attempts: 5
Indicate if each of the following is an expression or an equation:
· 4x+34x+3
· x2−16x−4×2-16x-4
· (x−4)(x+3)=6(x-4)(x+3)=6
· (x−4)(x−3)(x-4)(x-3)
· 4y=34y=3
#10 Points possible: 9. Total attempts: 5
The percentage of Americans who are retired has been increasing over the last decade. This is causing some concern because health care, social security, and other costs will be the responsibility of a smaller group of people. That is, as the percentage of retired people increases, the percentage of working age people decreases. The following model predicts the percentage of retired people based on demographic data1:
R=t873.36−2.15R=t873.36-2.15
where R is the percentage (as a decimal) of Americans who are retired in the year t (for example, the year 1995 would be t = 1995). Use this model to complete the table below by solving for the year.
Year | % of Retired People |
10% | |
15% | |
20% |
#11 Points possible: 5. Total attempts: 5
Andy’s house is on a large lot. He got 100 yards of chain-link fence on sale. He wants to use all of the material to completely enclose a rectangular area in his backyard. He wants to make the fenced area 60 feet wide and as long as possible. What is the longest length possible for the sides?
ft
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