Problem 67 - College Algebra

Question

Anthropologists and forensic scientists classify skulls using $$\frac{L+60 W}{L}-\frac{L-40 W}{L}$$ where \(L\) is the skull's length and \(W\) is its width a. Express the classification as a single rational expression. b. If the value of the rational expression in part (a) is less than \(75,\) a skull is classified as long. A medium skull has a value between 75 and \(80,\) and a round skull has a value over \(80 .\) Use your rational expression from part (a) to classify a skull that is 5 inches wide and 6 inches long.

Step-by-Step Solution

Answer

The simplified rational expression is \(\frac{100W}{L}\). Given the measurements of a skull that is 5 inches wide and 6 inches long, the skull would be classified as round because the calculated value of 83.33 exceeds 80.

Step 1: Simplify Rational Expression

The given rational expression is \(\frac{L+60 W}{L}-\frac{L-40 W}{L}\). First, find a common denominator to subtract these two fractions. Since \(L\) is the denominator for both fractions, this simplifies the process: \(\frac{L+60 W - (L-40 W)}{L} = \frac{L+60W-L+40W}{L} = \frac{100W}{L}\).

Step 2: Classify Skull Length

Following the classifications given in the problem, substitute the given measurements of width and length into the rational expression: \(\frac{100*5}{6}\). Simplify the expression to get a numerical value: \(\frac{500}{6} = 83.33\).

Step 3: Interpret Result

The resulting value is 83.33. Given the classifications in the problem, this numerical value represents a round skull. As the value is greater than 80, the skull is classified as round.

Key Concepts

Simplifying Rational Expressions

Simplifying rational expressions is just like simplifying regular fractions. Imagine fractions being expressions with letters and numbers. Here, we work to simplify them just as we would with numbers. The key is to manage their numerators and denominators skillfully.
In the given exercise, you're working with a rational expression: \( \frac{L+60 W}{L} - \frac{L-40 W}{L} \). Each fraction shares the same denominator, \(L\). This means you can subtract the numerators directly: \( L + 60W \) and \( L - 40W \).
Here's the simplified process:

• Subtract the second numerator from the first: \((L + 60W) - (L - 40W)\).
• Distribute the negative sign: \(L + 60W - L + 40W\).
• The \(L\) terms cancel each other out: \(60W + 40W = 100W\).

Now we have \(\frac{100W}{L}\), a much simpler expression! This makes calculations easier down the road.

Rational Expressions in Anthropology

In anthropology, rational expressions can be a helpful way to quantify and classify physical features, such as skull dimensions. By analyzing features like skull length and width, experts can use math to categorize these characteristics.
The rational expression in the exercise, \(\frac{100W}{L}\), allows anthropologists to systematically distinguish between different types of skulls. Here's how it works:

• "\(W\)" stands for skull width, and "\(L\)" for skull length.
• The result of \(\frac{100W}{L}\) gives a dimension ratio that helps in classifying skull shapes.
• This ratio can then be compared against predefined thresholds to determine if a skull is long, medium, or round.

Using these expressions means that scientists can be consistent and objective in their assessments, improving the way they interpret physical anthropological data.

Classification of Skulls

Classifying skulls based on their shape involves comparing a calculated value against set benchmarks. After simplifying the rational expression to \(\frac{100W}{L}\), you use this formula to determine the skull shape classification.
Given the thresholds provided in the problem:

• A skull is considered **long** if the result is less than 75.
• A skull is **medium** if the result is between 75 and 80.
• An **round** skull has a result over 80.

For example, using the values \(W = 5\) and \(L = 6\) in the expression \(\frac{100W}{L}\):

• Substituting in gives \(\frac{100 \times 5}{6} = \frac{500}{6}\).
• This simplifies to approximately 83.33.

Since 83.33 is greater than 80, the skull is classified as round. Understanding this mathematical model helps students see how quantifiable measurements connect to anthropological theory.