Question

Use the two rule-of-thumb approaches (by mass and by cost) to estimate the energy investment in a laptop computer, whose mass is \(1.4 \mathrm{~kg}\) and cost is \(\$ 1,300\). If the estimates are strikingly different, which do you suspect is more representative of the truth?

Short answer

Answer: By mass, the estimated energy investment is approximately 8.4 million joules. By cost, the estimated energy investment is approximately 5.2 billion joules. The energy investment by cost may be a more accurate representation. However, keep in mind that both estimates are rule-of-thumb approaches and actual values might vary.

Step-by-step solution

Calculate energy investment by mass

To estimate the energy investment by mass, we'll use the following rule-of-thumb: 1 kg of mass requires an energy investment of \(6\times10^{6} J\). Given the laptop computer has a mass of 1.4 kg, we'll multiply this value by the energy investment per kg. Energy investment by mass (in joules) = \(1.4 \mathrm{~kg} \times 6\times10^{6} J/\mathrm{kg}\)

Calculate energy investment by cost

To estimate the energy investment by cost, we'll use the following rule-of-thumb: 1 USD of cost requires an energy investment of \(4\times10^{6} J\). Given the laptop computer has a cost of 1300 USD, we'll multiply this value by the energy investment per USD. Energy investment by cost (in joules) = \(\$ 1,300 \times 4\times10^{6} J/\$ \)

Compare the two estimates

Next, we will calculate the energy investment values for both methods and compare the results. Energy investment by mass (in joules) = \(1.4 \times 6\times10^{6} J = 8.4\times10^{6} J\) Energy investment by cost (in joules) = \(1300 \times 4\times10^{6} J = 5.2\times10^{9} J\)

Determine which estimate is more representative of the reality

The energy investment by mass is 8.4 million joules, whereas the energy investment by cost is 5.2 billion joules. There is a significant difference between the two estimates. It suggests that looking at the cost of the laptop computer might give a more accurate representation of the energy investment required since it takes into account various components' production and transportation costs. However, it's important to remember that both are rule-of-thumb approaches, and actual energy investment values might vary.

Key concepts

Energy by Mass

When determining energy investment by mass, we utilize a straightforward rule-of-thumb. This method provides a rough estimate of the energy needed based solely on the weight of an item. This approximation is pretty handy since many people first think of an object's physical form. Here's how it works:

• It is generally assumed that each kilogram of an item requires approximately \(6 \times 10^6 \) joules for its production and delivery.
• Let's consider a laptop with a mass of 1.4 kg.
• We multiply the laptop's mass by the energy required per kg: \(1.4 \times 6 \times 10^6 = 8.4 \times 10^6 \) joules.

This tells us that roughly 8.4 million joules are needed for the laptop when viewed through this lens. While simple, this method doesn't account for material specifics or production complexities.

Energy by Cost

Looking at the energy investment by cost uses another rule-of-thumb. This approach takes into account the monetary value of an item, assuming that higher costs reflect more complex production and, therefore, greater energy expenditure. Here's a breakdown of this estimation:

• For each US dollar spent, an estimated \(4 \times 10^6 \) joules of energy is considered invested.
• The specific laptop in our example costs $1,300.
• To find the energy investment: multiply the cost by the energy per dollar: \(1,300 \times 4 \times 10^6 = 5.2 \times 10^9 \) joules.

Through this perspective, we estimate the laptop requires a whopping 5.2 billion joules! This method usually gives a higher energy estimate since it inherently includes many hidden energy costs like logistics and technological advancements reflected in price.

Rule-of-Thumb Approach

The rule-of-thumb approach is all about using simple, quick calculations to estimate complex problems. Both the energy by mass and energy by cost are this type of estimation. Here's why they can be both practical and limited:

• They offer a rapid way to approximate energy investments using typical values. This can be tremendously useful in comparisons, making informed guesses without intricate data.
• However, they overlook item-specific details. For example, high-tech materials may need more energy, or an item might be priced higher due to brand factors rather than energy input.
• In this case, the significant discrepancy between our energy estimates (8.4 million vs. 5.2 billion joules) emphasizes their roughness and caution needed when interpreting results.

These approaches serve as supplementary tools, which can guide decisions or provide insight for more detailed analyses.