Problem 22 - College Algebra

Question

Use the four-step procedure for solving variation problems given on page 424 to solve. An object's weight on the moon, \(M,\) varies directly as its weight on Earth, \(E .\) Neil Armstrong, the first person to step on the moon on July \(20,1969,\) weighed 360 pounds on Earth (with all of his equipment on) and 60 pounds on the moon. What is the moon weight of a person who weighs 186 pounds on Earth?

Step-by-Step Solution

Answer

The moon weight of a person who weighs 186 pounds on Earth is approximately 31 pounds

Step 1: Write the variation equation

The variation equation based on the problem is \(M = kE\) where \(M\) is the weight on the moon, \(E\) is the weight on Earth and \(k\) is the constant of variation.

Step 2: Substitute given values and solve for k

From the problem, Neil Armstrong's weight on Earth (\(E\)) is 360 pounds and his weight on the moon (\(M\)) is 60 pounds. Substituting these values into the equation gives us \(60 = k * 360\). Solve for \(k\) by dividing both sides by 360. The result is \(k = 60/360 = 0.1667\).

Step 3: Substitute value of k in the equation

After finding the value of \(k\), the initial equation \(M = kE\) now becomes \(M = 0.1667E\). This is the equation we'll use to find the weight of the person on the moon who weighs 186 pounds on Earth.

Step 4: Substitute the new value in the equation

Now, if a person weights 186 pounds on the Earth (\(E = 186\)), substitute this value into the equation gives \(M = 0.1667 * 186\). Solving this equation yields a moon weight of approximately 31 pounds.

Key Concepts

Direct Variation

In understanding direct variation, imagine a relationship between two quantities where as one quantity increases, the other increases at a constant rate. This is akin to saying, 'the faster you walk, the more ground you'll cover in the same amount of time.' Mathematically, this relationship is expressed as \( y = kx \), where \( y \) varies directly with \( x \) and \( k \) is a non-zero constant known as the constant of variation or proportionality constant.

In the exercise about the astronaut's weight, the moon weight \( (M) \) and earth weight \( (E) \) exhibit direct variation. They are connected by a consistent ratio, which means if you know one's weight on Earth, you can easily determine their weight on the moon by using the proportionality constant \( k \), illustrating the essence of direct variation in practical scenarios.

Constant of Variation

Think of the constant of variation as the factor that ‘glues’ together two varying quantities in a direct variation. It remains unchanged regardless of the values of the variables it connects. When we say \( y = kx \), the constant \( k \) is this 'glue.' If you know the constant of variation, you can predict one variable if the other is known.

As seen in the astronaut example, we found the constant of variation (\( k = 0.1667 \)) by using the known weights of the astronaut on Earth and the moon. Once determined, this constant allows us to calculate the moon weight for any given Earth weight. This step is pivotal as it establishes the specific direct relationship between Earth and moon weights for all objects, not just the astronaut.

Proportional Relationships

Proportional relationships are the heart of understanding direct variation. They describe how two quantities increase or decrease at the same rate. You can visualize it as two dancers moving in sync, each step of one dancer is matched by a step of the other. Mathematically, when we say two variables are proportional, we imply that their ratio is constant.

Using our moon weight equation, the ratio of moon weight to Earth weight is constant for all objects, not just unique instances. That's why when the astronaut’s weight was used to determine the constant of variation, it could then be applied universally to find any other person's moon weight, assuming the proportional relationship holds true. This fundamental concept underpins much of algebra and helps students solve a wide variety of problems by establishing predictable, consistent relationships between variables.