Problem 66 - College Algebra with Corequisite Support

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: \( 6 - \frac{2}{\sqrt{t}} \text{ m/s}^2 \).

Step 1: Identify the given expression

The given expression for the acceleration of the car is: \[ 6 - \frac{\sqrt{4}}{\sqrt{t}} \text{ m/s}^2 \] We need to simplify this expression.

Step 2: Simplify the square root in the numerator

The numerator of the fraction is \( \sqrt{4} \). Since the square root of 4 is 2, we can simplify it as:\[ \sqrt{4} = 2 \] Substitute this back into the expression.

Step 3: Substitute the simplified numerator

The expression becomes: \[ 6 - \frac{2}{\sqrt{t}} \text{ m/s}^2 \] Now, this is the simplified expression for the acceleration as a function of time \( t \).

Key Concepts

Square Roots

Square roots are a crucial part of many mathematical operations. The square root of a number is a value that, when multiplied by itself, gives the original number. For example, the square root of 4 is 2 because 2 times 2 equals 4. This is mathematically represented as \( \sqrt{4} = 2 \).

When simplifying expressions involving square roots, it's essential to know these basic calculations. For instance, in our exercise, we have \( \sqrt{4} \) in the numerator. By knowing the square root of 4, we easily reduce the fraction to \( \frac{2}{\sqrt{t}} \).

It's often helpful to memorize the square roots of perfect squares (e.g., 1, 4, 9, 16) for quicker simplifications in problems.

• \( \sqrt{1} = 1 \)
• \( \sqrt{9} = 3 \)
• \( \sqrt{16} = 4 \)
• \( \sqrt{25} = 5 \)... and so on.

Understanding these basics can make dealing with more complex expressions easier and more intuitive.

Acceleration

Acceleration is the rate at which an object's velocity changes over time. It's often measured in meters per second squared (m/s²). In our problem, we have an expression representing the car's acceleration as it starts from rest.

The original expression is \[ 6 - \frac{\sqrt{4}}{\sqrt{t}} \text{ m/s}^2 \].
Once simplified by calculating the square root of 4, it becomes:

\[ 6 - \frac{2}{\sqrt{t}} \text{ m/s}^2 \].

• The constant 6 in the expression signifies that without the influence of time, the car has a baseline acceleration.
• The term \( -\frac{2}{\sqrt{t}} \) suggests a decrease in acceleration as time progresses.

Understanding acceleration helps in predicting how fast an object changes its speed and is fundamental in physics and engineering calculations.

Mathematical Functions

Mathematical functions provide a way to express relationships between two or more variables, often describing patterns or rules within a problem. Functions like the one in our exercise give us an acceleration value based on the variable time, \( t \).

The given function for acceleration is:\[ a(t) = 6 - \frac{2}{\sqrt{t}} \]where \( a(t) \) is a function of time that calculates acceleration at any given time \( t \).

Functions can often involve different operations like addition, subtraction, multiplication, division, and even functions within functions.

• Here, we see how square root functions are used within the primary function.
• This nested structure allows for more complex relationships to be expressed in a compact form.

By understanding mathematical functions, we can model real-world situations and solve practical problems efficiently.