Question

A submarine currently operating at 100 feet below the surface of the ocean begins to dive to increased depths at a rate of 30 feet per minute. Which of the following equations represents the submarine's depth below the surface, \(d\), in feet, \(m\) minutes after beginning to dive? A) \(d=100+30 \mathrm{~m}\) B) \(d=100-30 \mathrm{~m}\) C) \(d=100-30\) D) \(d=100 m-30\)

Short answer

The correct equation that represents the submarine's depth below the surface is option A: \(d = 100 + 30m\).

Step-by-step solution

Identify initial depth

The submarine is initially 100 feet below the surface. This will be the starting point for the equation.

Identify rate of descent

The submarine descends at a rate of 30 feet per minute. This means that the depth below the surface will increase by 30 feet for every minute.

Write depth as a function of time

Since the depth increases as time passes, we can write an equation that shows the relationship between depth \(d\) and minutes \(m\). We start with the initial depth and add the rate of descent multiplied by the minutes passed: \(d = 100 + 30m\)

Choose the correct equation

Based on our analysis, the correct equation that represents the submarine's depth below the surface is option A: \(d = 100 + 30m\).

Key concepts

Linear Equations

Solving problems involving linear equations is a fundamental skill in algebra. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations are used to represent a continuous relationship between quantities, with one variable typically dependent on another.

The hallmark of a linear equation is its graph, which is a straight line on a coordinate plane. In the case of the submarine diving into the ocean, we're looking at a real-world application of linear equations, where the depth of the submarine is directly proportional to the time it spends diving.

To write a linear equation for the given problem, we start with the initial value (the y-intercept, in this case, the submarine's initial depth of 100 feet) and incorporate the rate of change (the slope, here 30 feet per minute as the submarine descends). This gives us the linear equation that effectively predicts the submarine's depth for any minute after the dive begins: \(d = 100 + 30m\).

Algebraic Expressions

An algebraic expression is a mathematical phrase that can contain ordinary numbers, variables (like \(m\) or \(x\)), and operators (such as add, subtract, multiply, and divide). The purpose of an algebraic expression is to denote a rule for the relationship between quantities.

In our submarine scenario, the algebraic expression is \(100 + 30m\), which neatly packages the rule for finding the depth at any given minute. The initial depth, 100 feet, is a constant, reflecting a value that doesn't change as the submarine continues to dive. The term \(30m\) represents the change in depth, with \(m\) being a variable that stands for the minutes passed.

The power of algebraic expressions lies in their ability to generalize rules for numerical relationships, allowing the same expression to be used for numerous values of the variables involved. By understanding algebraic expressions, students can find solutions to a wide variety of problems.

Rate of Change

In mathematics, the rate of change is a concept that measures how a quantity changes with respect to another quantity, often time. In our example, the rate of change is the submarine's descent speed, 30 feet per minute. It represents the slope of the linear equation and is a key concept in understanding how to create equations from real-world scenarios.

The rate of change is what connects linear equations to real-life situations, as it measures how quickly something is moving or changing. In this case, the positive rate of change means that the depth is increasing over time as the submarine dives deeper. It's crucial for students to grasp that the rate of change in the context of linear equations is constant, which is why the graph of a linear equation is always a straight line.

Understanding the rate of change not only helps with solving math problems but also with interpreting those problems in a practical, real-world context. This is incredibly useful in fields such as physics, economics, and various branches of engineering where rates are constantly analyzed and utilized.