Problem 18 - Algebra 2

Question

When a sound travels through water, the distance \(y\) in meters that the sound travels in \(x\) seconds is given by the equation \(y=1440 x .\) How far does a sound travel underwater in 5 seconds?

Step-by-Step Solution

Answer

The sound travels 7200 meters underwater in 5 seconds.

Step 1: Identify the Given Information

We start by identifying the given equation and the values for the variables in the problem. The equation is \(y = 1440x\), where \(y\) is the distance traveled in meters, and \(x\) is the time in seconds. We are given \(x = 5\) seconds.

Step 2: Substitute the Value of x

We substitute the given \(x\) value into the distance equation. Replace \(x\) with 5 in the equation: \(y = 1440 \times 5\).

Step 3: Calculate the Distance

Now, perform the multiplication to find \(y\). Compute \(1440 \times 5 = 7200\). Thus the distance \(y\) is 7200 meters.

Key Concepts

Substitution

When we talk about substitution in mathematics, we refer to the process of replacing a variable with a specific numerical value. This is a vital step in solving many algebraic equations. By substituting, we focus on evaluating the expression to solve the problem at hand. In our specific exercise, the distance formula is given as \(y = 1440x\). Here, \(x\) represents the time in seconds during which the sound travels through water. The problem asks us to find the distance traveled in 5 seconds, so we substitute \(x = 5\) directly into the formula.

• This substitution modifies the equation to \(y = 1440 \times 5\).
• It's crucial because it turns a general equation into one that can deliver a specific numerical result.

This step sets the stage for precise calculations which leads us to the solution. Remember, whenever you see a variable, substitution allows you to test out various values or solve for specific conditions.

Calculation

Calculation is the process of computing an answer using mathematical operations. In our problem, after substituting the value of \(x\) with 5, the calculation involves performing the operation indicated in the equation: \(y = 1440 \times 5\). This step is essential as it leads directly to the answer by applying arithmetic operations:

• First, arrange the numbers: \(1440\) and \(5\).
• Next, perform the multiplication to solve for \(y\).
• The result is \(7200\), indicating that the sound travels 7200 meters in 5 seconds.

By focusing on the arithmetic calculation, you ensure accuracy and understanding of how the equation derives its result. It's vital to check your work during this step to confirm the numbers are correctly used and the operations properly performed.

Multiplication

Multiplication is a fundamental mathematical operation that represents the concept of adding a number to itself a certain number of times. In the context of our exercise, multiplication helps us determine the distance a sound wave travels in a given time.To understand the operation, here is how it unfolds in our problem:

• The equation \(y = 1440 \times 5\) tells us to multiply 1440, the speed in meters per second, by 5, the time in seconds.
• This multiplication \(1440 \times 5 = 7200\) calculates the total distance, as it combines uniform speed with the time duration.

By mastering multiplication, you can compute distances or any other measurements that rely on a constant rate or repeated addition. This skill forms the bedrock of many algebraic solutions and is critical in analyzing and solving distance-time related problems efficiently.