Question

Calculate Fuel A spacecraft carries \(30 \mathrm{~kg}\) of hydrazine fuel and uses and average of \(500 \mathrm{~g} / \mathrm{y}\). How many years could this fuel last?

Short answer

The fuel will last for 60 years.

Step-by-step solution

Understanding the Problem

The spacecraft has 30 kg of hydrazine fuel and uses an average of 500 g per year. We need to determine how many years this fuel will last before running out.

Convert Units

Convert the total fuel from kilograms to grams since the usage is given in grams/year. We know that 1 kg = 1000 g, so 30 kg = 30,000 g.

Calculate Duration

Divide the total amount of fuel by the average usage per year to find out how many years the fuel will last. Use the formula: \[ \text{Years of fuel} = \frac{\text{Total fuel (g)}}{\text{Fuel usage per year (g)}} \] Here, \( \text{Total fuel} = 30,000 \text{ g} \) and \( \text{Fuel usage per year} = 500 \text{ g} \).

Perform Calculation

Perform the division: \( \frac{30,000 \text{ g}}{500 \text{ g/year}} = 60 \text{ years} \). Thus, the fuel will last for 60 years.

Key concepts

Unit Conversion

Understanding unit conversion is essential when dealing with quantities measured in different units. In this exercise, we have a spacecraft carrying 30 kg of fuel, but the fuel consumption rate is given in grams per year. To accurately calculate how long the fuel will last, we need to express both quantities in the same units.

By knowing the conversion factor, we can switch between these units seamlessly. Since 1 kilogram equals 1000 grams, converting the 30 kg of fuel to grams involves multiplying 30 by 1000, resulting in 30,000 grams. This step allows us to match the units of the fuel capacity with the consumption rate, ensuring a correct and straightforward calculation of how long the fuel will last.

Unit conversion helps avoid mistakes and confusion when performing calculations with different units. It is a practical skill often used in various fields, especially in sciences and engineering.

Mathematical Modeling

Mathematical modeling involves representing real-world scenarios in mathematical terms, allowing us to analyze and solve problems effectively. In this context, the problem involves a straightforward mathematical model. We are tasked with determining how long 30,000 grams of fuel will last, given a consumption rate of 500 grams per year.

The model uses a simple division formula:

• \( \text{Years of fuel} = \frac{\text{Total fuel (g)}}{\text{Fuel usage per year (g)}} \)

This formula enables us to input the known values and find the unknown duration, which is critical for planning and decision-making in resource management.

The beauty of mathematical modeling is its ability to simplify complex problems, enabling us to find solutions rapidly. This is particularly useful in fields like engineering, where precise calculations are necessary to ensure the success of missions or projects.

Resource Management

Resource management refers to the strategic use of resources to maximize efficiency and efficacy. In this case, effective management of the spacecraft's fuel is crucial for ensuring long missions without exhausting supplies prematurely.

By calculating how many years the fuel will last, you can make informed decisions about the mission's duration and potential resupply needs. This avoids unnecessary risk to the mission, ensuring that there is always a plan in place for maintaining critical operations.

• Calculating fuel duration before a mission is an essential component of resource management.
• Forecasting allows for planning and taking corrective measures if resources run low earlier than expected.

Understanding and implementing rigorous resource management strategies can lead to better outcomes, extending mission capabilities and reducing costs.