Problem 76 - Introductory Algebra for College Students

Question

The temperature, in degrees Fahrenheit, of a dessert placed in a freezer for \(t\) hours is modeled by $$ \frac{t+30}{t^{2}+4 t+1}-\frac{t-50}{t^{2}+4 t+1} $$ a. Express the temperature as a single rational expression. b. Use your rational expression from part (a) to find the temperature of the dessert, to the nearest hundredth of a degree, after 1 hour and after 2 hours.

Step-by-Step Solution

Answer

a. The temperature of the dessert as a single rational expression is \(\frac{80}{t^{2}+4 t+1}\) Fahrenheit. b. After 1 hour, the temperature of the dessert is about 13.33 degrees Fahrenheit, and after 2 hours, it's approximately 5.71 degrees Fahrenheit.

Step 1: Simplify the rational expression

The rational expression given is \(\frac{t+30}{t^{2}+4 t+1}-\frac{t-50}{t^{2}+4 t+1}\). Since both fractions have the same denominator, they can be combined into a single fraction by subtracting the numerators. The combined expression becomes \(\frac{(t+30) - (t-50)}{t^{2}+4 t+1}\). Simplifying the numerator gives \(\frac{80}{t^{2}+4 t+1}\)

Step 2: Find the temperature after 1 hour

Plug \(t = 1\) into the simplified rational expression. This yields \(\frac{80}{1^{2}+4 \cdot 1+1} = \frac{80}{6} = 13.33\) degrees Fahrenheit.

Step 3: Find the temperature after 2 hours

Plug \(t = 2\) into the simplified rational expression. This produces \(\frac{80}{2^{2}+4 \cdot 2+1} = \frac{80}{14} = 5.71\) degrees Fahrenheit.

Key Concepts

Algebra Simplification

Simplifying algebraic expressions is an essential tool in mathematics, making calculations easier and more straightforward. When dealing with rational expressions, such as the one given in the temperature modeling problem, simplification can help combine several terms into one concise expression.

Here, we have two rational expressions with a common denominator: \(\frac{t+30}{t^{2}+4t+1}\) and \(\frac{t-50}{t^{2}+4t+1}\). The beauty of having identical denominators is that it allows us to subtract these fractions easily by just focusing on the numerators.

• Both fractions are written over the same base, \(t^{2}+4t+1\).
• The numerators \((t+30)\) and \((t-50)\) are subtracted: \((t+30)-(t-50)\).
• This subtraction simplifies to \(80\) because the \(t\) terms cancel out, leaving \(30 - (-50)\).

Ultimately, we transform the expression into the much simpler form \(\frac{80}{t^{2}+4t+1}\), making further calculations or interpretations easier to handle.

Temperature Modeling

In the context of temperature modeling, rational expressions like the one given are used to describe how temperatures change over time in specific scenarios. By expressing temperature as a function of time, we can predict future temperatures and understand past behaviors.

In this problem, the temperature of a dessert in a freezer over time \(t\) is expressed by the rational expression \(\frac{80}{t^{2}+4t+1}\). This model helps us determine the temperature at any given time \(t\) by plugging in values for \(t\), allowing us to see how the temperature decreases as the dessert cools.

Such models are especially useful in scientific and culinary fields where understanding the rate of cooling is crucial. They help construct mathematical representations that mimic real-life conditions. In this case:

• At \(t = 1\), the temperature is approximately \(13.33\) degrees Fahrenheit.
• At \(t = 2\), it decreases further to about \(5.71\) degrees Fahrenheit.

These calculations show the dessert's rapid cooling as it stays in the freezer longer.

Fraction Subtraction

Fraction subtraction is a basic, yet vital, operation in algebra that aids in solving more complex equations involving rational expressions. When fractions have the same denominator, like in our initial problem, it simplifies the process significantly.

Subtracting fractions with a common denominator follows straightforward steps:

• Write both fractions with the same denominator (e.g., \(t^{2}+4t+1\)).
• Subtract the numerators directly: \((t+30) - (t-50)\).
• Simplify the resulting numerator to find \(80\).

The subtraction process results in a simple expression \(\frac{80}{t^{2}+4t+1}\), which combines the two original fractions into one rational expression. With this form, we can quickly proceed to evaluate the fraction at specific values of \(t\), like when the time is 1 hour or 2 hours. This straightforward method of handling common denominators is a crucial skill for simplifying algebraic expressions.