Problem 112 - College Algebra with Modeling and Visualization

Question

The weight of an object on Earth is directly proportional to the weight of an object on Mars. If a 25 -pound object on Earth weighs 10 pounds on Mars, how much would a 195 -pound astronaut weigh on Mars?

Step-by-Step Solution

Answer

The astronaut would weigh 78 pounds on Mars.

Step 1: Understand the Proportional Relationship

The problem states that weight on Earth is directly proportional to weight on Mars. This means we can express this relationship as \( W_m = k \times W_e \), where \( W_m \) is the weight on Mars, \( W_e \) is the weight on Earth, and \( k \) is the proportionality constant.

Step 2: Find the Proportionality Constant

To find the constant \( k \), use the given information: a 25-pound object on Earth weighs 10 pounds on Mars. Substitute these values into the equation: \( 10 = k \times 25 \). Solve for \( k \): \[k = \frac{10}{25} = \frac{2}{5} = 0.4\]

Step 3: Calculate the Astronaut's Weight on Mars

Use the constant \( k = 0.4 \) to find the weight of a 195-pound astronaut on Mars. Substitute 195 for \( W_e \): \[W_m = 0.4 \times 195 = 78\]

Step 4: Verify the Calculation

Double-check calculations to ensure accuracy: \( 195 \times 0.4 = 78 \).Calculations are correct, thus confirming the Mars weight as 78 pounds.

Key Concepts

Weight Conversion

Weight conversion involves the process of translating the weight of an object from one measurement system to another, or in different gravitational fields. When discussing weight conversion in the context of planetary weights, we're referring to how an object's weight changes due to the gravitational forces on different planets. For instance, an object might weigh 25 pounds on Earth, but under Mars' weaker gravity, its weight converts to just 10 pounds.

To convert weights between such different gravitational conditions, first, identify the proportionality constant. This includes comparing a known weight on one planet to the alternate condition. With this constant, any weight can be recalculated across other planets. This approach helps in better understanding the differences in gravitational pull, which can affect how one experiences weight.

Planetary Weights

When we talk about planetary weights, we are focusing on how gravity varies by planet, affecting how heavy things feel or physically weigh. Each planet in our solar system exerts its own specific gravitational force, which interacts with an object's mass to produce weight.

An object might weigh significantly less or more when taken from one planet’s surface to another. For example, the gravitational force on Mars is about 38% of that on Earth. This is why a 25-pound object on Earth translates to a weight of only 10 pounds on Mars.

• To calculate these weights, use the concept of direct proportionality, relating Earth's weight to Mars' weight with a fixed proportionality constant.
• This allows any Earth-based weight to be converted to Mars by multiplying by 0.4, as calculated with known weights, such as from the exercise.

This conversion helps scientists and engineers predict and plan for extraterrestrial missions by understanding how much equipment and even astronauts will weigh on different planets.

Algebraic Equations

Algebraic equations are mathematical statements that show the equality of two expressions. The core idea is using known information to find unknown values. In our example involving planetary weights, algebra helps define and solve for weights based on known conditions.

To determine the astronaut's weight on Mars, we first establish a general equation: \( W_m = k \times W_e \), where \(W_m\) is the weight on Mars, \(W_e\) is the weight on Earth, and \(k\) is the proportionality constant.

Given the specific conditions of a 25-pound weight on Earth translating to 10 pounds on Mars, substitute these values into the equation to solve for \(k\): \[k = \frac{10}{25} = 0.4\]. Once \(k\) is determined, apply it to any Earth's weight to find the respective Mars weight by plugging in different values for \(W_e\).

• Understanding and solving algebraic equations form the base for converting planetary weights, allowing us to predict how much different Earth objects will weigh more accurately under another planet's gravity.

This methodology offers broader applications than astronomy, simplifying problems into approachable formulas across various fields.