## Putting It Together: Solving Equations and Inequalities

At the start of this module, you were presented with a formula commonly used by forensic scientists to calculate blood alcohol content:

$B=-0.015t +\left(\frac{2.84N}{Wg}\right)$

where

• B = percentage of BAC
• t = number of hours since the first drink
• 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

Remember Joan? When we left her, she was at a party. She had three drinks and then wanted to leave. This was worrisome. We posed the following questions:

1. Where would she fall on the table of the progressive effects of alcohol after 1.5 hours?
2. Would she be within the legal limit to drive after this amount of time?

The table showing the progressive effects of alcohol appears at the bottom of the page.

The information needed to answer the first question includes BAC, N, W, t and g.  Of these, we know N, W, t, and g—we will need to solve for BAC.

• N = 3 standard drinks
• W = 135 pounds
• t = 1.5 hours
• g = 0.55

Using these values, we can calculate Joan’s BAC, which is the first step in answering question 1.

$\begin{array}{l}\text{B} = -0.015t +\left(\frac{2.84N}{Wg}\right)\\\text{B} = -0.015(1.5) +\left(\frac{2.84(3)}{(135)(0.55)}\right)\\\text{B} = -0.0225 +\left(\frac{8.52}{(74.25)}\right)\\\text{B} = -0.0225 +\left(0.1147\right)\\\text{B} = 0.092\\\end{array}$

The units of B are a percentage, so in this scenario after 1.5 hours and after consuming three standard drinks, Joan’s BAC is 0.092%.

How is Joan doing? From the table below, we can predict that she’s experiencing blunted feelings, reduced sensitivity to pain, euphoria, disinhibition, and extroversion. In addition, her reasoning, depth perception, peripheral vision, and glare recovery are probably impaired.

Question 2 asks whether she would be within the legal limit to drive.  If we assume that the legal limit is 0.08%, then NO, she would not be within the legal limit to drive. One of her friends should take her car keys and help her get home.

Progressive effects of alcohol
BAC (% by vol.) Behavior Impairment
0.001–0.029
• Average individual appears normal
• Subtle effects that can be detected with special tests
0.030–0.059
• Mild euphoria
• Relaxation
• Joyousness
• Talkativeness
• Decreased inhibition
• Concentration
0.060–0.099
• Blunted feelings
• Reduced sensitivity to pain
• Euphoria
• Disinhibition
• Extroversion
• Reasoning
• Depth perception
• Peripheral vision
• Glare recovery
0.100–0.199
• Over-expression
• Boisterousness
• Possibility of nausea and vomiting
• Reflexes
• Reaction time
• Gross motor control
• Staggering
• Slurred speech
• Temporary erectile dysfunction
0.200–0.299
• Nausea
• Vomiting
• Emotional swings
• Partial loss of understanding
• Impaired sensations
• Decreased libido
• Possibility of stupor
• Severe motor impairment
• Loss of consciousness
• Memory blackout
0.300–0.399
• Stupor
• Central nervous system depression
• Loss of understanding
• Lapses in and out of consciousness
• Low possibility of death
• Breathing
• Dysequilibrium
• Heart rate
0.400–0.500
• Severe central nervous system depression
• Coma
• Possibility of death
• Breathing
• Heart rate
• Positional Alcohol Nystagmus
>0.50
• High risk of poisoning
• High possibility of death
• Life