Problem 28 - College Algebra

Question

Use the four-step procedure for solving variation problems given on page 424 to solve. The water temperature of the Pacific Ocean varies inversely as the water's depth. At a depth of 1000 meters, the water temperature is \(4.4^{\circ}\) Celsius. What is the water temperature at a depth of 5000 meters?

Step-by-Step Solution

Answer

The water temperature at a depth of 5000 meters is \(0.88^{\circ}\) Celsius.

Step 1: Determine the constant of variation

An inverse variation relationship can be represented by the formula \(xy = k\) where x and y are the variables and k is the constant of variation. Let's plug in the given values of depth (x) as 1000 meters and temperature (y) as \(4.4^{\circ}\) Celsius. Using the formula, we get \(1000*4.4 = k\), hence \(k = 4400\).

Step 2: Use the constant of variation to find the temperature at 5000 meters

Now that we have the constant of variation, we can use it to find the temperature at 5000 meters. Plugging the values into the formula \(xy = k\), where x = 5000 (the new depth) and k = 4400 (the constant of variation). We now need to solve for y (the unknown temperature). Hence, the formula will be \(5000*y = 4400\). To solve for y, we divide both sides of the equation by 5000, getting \(y = 4400 / 5000\).

Step 3: Calculate the final temperature

Dividing 4400 by 5000 gives us \(y = 0.88\). This means that the temperature at a depth of 5000 meters is \(0.88^{\circ}\) Celsius.

Key Concepts

Constant of Variation

In algebra, the constant of variation is a crucial component when we address problems involving direct or inverse variation. It represents the consistent value that relates the two varying quantities. Particularly in inverse variation, the product of the two related quantities remains constant. In other words, if one variable increases, the other must decrease proportionally to maintain the constant product. For example, if depth and temperature of the ocean are inversely related, as the depth increases, the temperature decreases, but their product will always equal the constant of variation. Calculating this constant is typically the first step in solving an inverse variation problem.

Solving Variation Problems

To effectively resolve variation problems, understanding the relationship between the variables is important. The initial step generally involves identifying whether the variation is direct or inverse. Once known, you apply the corresponding algebraic formula. With inverse variation, the formula is usually expressed as \(xy = k\), where \(x\) and \(y\) are the inversely varying quantities and \(k\) is the constant of variation. After finding the constant, you can then use it to solve for the unknown variable when one of the other quantities changes. The process involves substituting the known values into the formula and using algebraic methods such as cross multiplication or division to solve for the unknown.

Direct and Inverse Variation

Understanding the differences between direct and inverse variation is a fundamental concept in algebra. In direct variation, two quantities increase or decrease together at a constant rate; this is expressed algebraically as \(y = kx\), where \(k\) is the constant of variation. Contrastingly, in inverse variation, as one quantity increases, the other decreases. Their product remains constant, expressed as \(xy = k\). Recognizing which type of variation is present in a problem is key to selecting the correct approach for solving it.

Algebraic Equations

Algebraic equations are mathematical statements that assert the equality of two expressions. They consist of variables and constants, and can represent a variety of relationships, such as rate, proportion, or variation. Solving algebraic equations involves finding the value(s) of the variable(s) that make the equation true. Techniques for solving equations can include various methods like adding or subtracting terms on both sides, multiplying or dividing by a number, and factoring. Mastery of algebraic equations is not only vital in mathematics but also in understanding real-world scenarios and other sciences.