Question

In extrapolating a \(2.3 \%\) growth rate in energy, we came to the absurd conclusion that we consume all the light from all the stars in the Milky Way galaxy within 2,500 years. How much longer would it take to energetically conquer 100 more "nearby" galaxies, assuming they are identical to our own?

Short answer

Answer: Approximately 920.59 years.

Step-by-step solution

Calculate the total energy consumption for the first 100 galaxies

Assuming each of the galaxies has the same amount of energy as the Milky Way, to consume the energy of 100 more galaxies, we need a total of 100 times the energy that is being consumed in the Milky Way after 2,500 years. We can use the formula for the geometric series to find the total energy consumed: \(S_n = \frac{a_1(1-r^n)}{1-r}\) where \(S_n\) represents the sum of the series, \(a_1\) is the first term, \(r\) is the common ratio, and \(n\) is the number of terms in the series. Here, we're considering each galaxy's energy as a term in the geometric series, so \(a_1 = 1\) (the energy consumed in the Milky Way after 2,500 years), the common ratio \(r=1 (100\%)\) and the number of terms \(n=101\) (1 Milky Way + 100 nearby galaxies). Plugging these values into the formula, we can compute the total energy consumed as: $S_{101} = \frac{1(1-(1)^{101})}{1-1}" Since r=1, the sum of the series becomes the total number of galaxies considered, which is 101.

Calculate the required time to consume the energy of 100 more galaxies

Now that we know the total energy that needs to be consumed, we can use the compound interest formula for annual growth to find the required time in years: \(A = P(1 + r)^t\) Where \(A\) is the final amount of energy consumed, \(P\) is the principal amount (initial energy of the Milky Way, which is considered as 1), \(r\) is the annual growth rate as a decimal (2.3% growth rate, so \(0.023\)), and \(t\) is the time in years. We know that \(A=101\), \(P=1\), and \(r=0.023\). We need to find the value of \(t\). \(101 = 1(1 + 0.023)^t\) We can rearrange the expression to solve for \(t\): \(t = \frac{\ln(\frac{A}{P})}{\ln(1+r)}\) \(t = \frac{\ln(\frac{101}{1})}{\ln(1+0.023)}\) Using a calculator to solve for \(t\), we get: \(t \approx 920.59\) years So, it would take approximately 920.59 years to energetically conquer 100 more "nearby" galaxies, assuming they are identical to our own.

Key concepts

Extrapolation of Energy Growth

The concept of extrapolation involves projecting current trends into the future to estimate an outcome. In the context of energy growth, if we notice that energy consumption is increasing by a certain percentage each year, we can extrapolate this trend to predict future energy needs.

Here's how it works: If we start with a certain amount of energy consumption and assume that it grows by a fixed percentage (like the 2.3% growth rate mentioned in our exercise), we can estimate how energy consumption will expand over time. The example given in the exercise reaches a seemingly absurd conclusion by showing how fast energy consumption could increase to a scale that's not sustainable or realistic, i.e., consuming all the light from the stars in our galaxy. By projecting beyond that, to 100 more galaxies, the exercise serves to highlight the limits of exponential growth in a finite universe.

Geometric Series in Energy Consumption

A geometric series is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. It's a mathematical concept that helps us understand scenarios like recurring growth or decline, which is exactly what happens with energy consumption.

For instance, if our energy consumption increases by a certain percentage annually, that's a geometric progression. The formula from the exercise, \(S_n = \frac{a_1(1-r^n)}{1-r}\), helps us sum up a series of such increments or contributions (from expanding energy usage across galaxies in this case). The exercise applies this when calculating the energy consumption for 101 galaxies assuming each increase is identical (a common ratio of 1), leading to the natural conclusion that the sum is simply the number of galaxies.

Compound Interest Formula for Energy Growth

The compound interest formula is not just for finance; it's widely applicable in any scenario that involves growth over periods, such as the growth of energy consumption. The formula \(A = P(1 + r)^t\) enables us to compute the final amount (\(A\)) based on the initial value (\(P\)), the growth rate (\(r\)), and the time passed (\(t\)).

In our exercise, we use this formula to determine how long it would take for energy consumption to grow to a level where it encompasses 101 galaxies. By rearranging the formula to isolate \(t\), we can solve for the time required given a particular growth rate—revealing the compounding effect of continuous growth over time and illustrating the timeline to reach unsustainable levels of consumption.