Problem 66 - College Algebra

Question

A car accelerates at a rate of \(6-\frac{\sqrt{4}}{\sqrt{t}} \mathrm{~m} / \mathrm{s}^{2}\) where \(t\) is the time in seconds after the car moves from rest. Simplify the expression.

Step-by-Step Solution

Answer

The simplified expression is \(6 - \frac{2\sqrt{t}}{t} \).

Step 1: Understand the Problem

We have the formula for acceleration, which depends on time \(t\): \(6 - \frac{\sqrt{4}}{\sqrt{t}}\). We are asked to simplify this expression.

Step 2: Simplify the Numerator

The numerator is \(\sqrt{4}\). We know that \(\sqrt{4} = 2\). So, our expression now is \(6 - \frac{2}{\sqrt{t}}\).

Step 3: Rationalize the Denominator

To simplify \(\frac{2}{\sqrt{t}}\), multiply the numerator and the denominator by \(\sqrt{t}\): \(\frac{2}{\sqrt{t}} \times \frac{\sqrt{t}}{\sqrt{t}} = \frac{2\sqrt{t}}{t}\).

Step 4: Simplify the Expression

Insert the simplified term back into the original expression: \(6 - \frac{2\sqrt{t}}{t}\). This is the simplified form of the acceleration.

Key Concepts

Square Roots

Square roots are a fundamental mathematical concept often encountered in algebra. A square root of a number is a value that, when multiplied by itself, gives the original number. For instance,

• The square root of 4 is 2 because 2 multiplied by 2 equals 4.
• Square roots can be written as \( \sqrt{...} \). For example, the square root of 4 is written as \( \sqrt{4} = 2 \).

Understanding square roots is crucial in algebra as they frequently appear in expressions and equations. When simplifying expressions involving square roots, like \( \frac{\sqrt{4}}{\sqrt{t}} \), it's essential to simplify each square root independently before proceeding to other operations. Here, simplifying \( \sqrt{4} \) to 2 helps make the expression easier to work with.

Rationalizing Denominators

Rationalizing denominators is the process of eliminating square roots or other irrational numbers from the denominator of a fraction. It's a common technique used to make expressions easier to understand and work with. To rationalize a denominator involving a square root:

• Multiply both the numerator and the denominator by a value that will make the denominator a rational number.
• For example, in the expression \( \frac{2}{\sqrt{t}} \), multiply both the numerator and the denominator by \( \sqrt{t} \) to get \( \frac{2\sqrt{t}}{t} \).

This process not only removes the square root from the denominator but also results in a simplified expression. Rationalization is a powerful algebraic technique used to maintain the clarity and accuracy of fractions.

Acceleration Formula

The acceleration formula expresses how an object's speed changes over time. In this case, the expression \(6 - \frac{2\sqrt{t}}{t}\) represents the car's acceleration depending on the variable \(t\), which is often time. Understanding how each term contributes:

• The number 6 represents constant acceleration.
• The term \(-\frac{2\sqrt{t}}{t}\) indicates the adjustment in acceleration over time.

When dealing with formulas related to motion, like acceleration, simplifying them can offer better insights into how variables interact. It allows easy evaluation and analysis of the physical phenomena depicted by the formulas.

Algebraic Fractions

Algebraic fractions are fractions where both the numerator and the denominator are algebraic expressions. Simplifying these types of fractions involves reducing them to their simplest form by factoring and canceling common terms if possible. Here's how you can work with algebraic fractions:

• Identify and simplify each component expression, like \( \sqrt{4} \), to make calculations easier.
• Use techniques like rationalization to manage complex fractions.

In the expression \(6 - \frac{2\sqrt{t}}{t}\) from our problem, we simplified it by first working with the square root and then rationalizing the fraction. This approach demonstrates effective methods in algebra to transform fractions into a more workable and understandable form.