## 1. Drake’s equation

In 1961 the scientist Francis Drake proposed the following formula to estimate the number of alien civilizations that can contact us

\[N = R_* \cdot f_\mathrm{p} \cdot n_\mathrm{e} \cdot f_\mathrm{l} \cdot f_\mathrm{i} \cdot f_\mathrm{c} \cdot L\]

where

- \(N\) is the number of civilizations in our galaxy with which communication might be possible
- \(R_{∗}\) is the average rate of star formation in our Galaxy
- \(f_{p}\) is the fraction of those stars that have planets
- \(n_{e}\) is the average number of planets that can potentially support life per star that has planets
- \(f_{l}\) is the fraction of planets that could support life that actually develop life at some point
- \(f_{i}\) is the fraction of planets with life that actually go on to develop civilizations
- \(f_{c}\) is the fraction of civilizations that develop a technology that releases detectable signs of their existence into space
- \(L\) is the length of time for which such civilizations release detectable signals into space

Drake made the following estimations

- \(R_{*}\) = 1 yr
^{-1}(1 star formed per year, on average) - \(f_{p}\) = 0.2 to 0.5 (1/5 to 1/2 of all stars will have planets)
- \(n_{e}\) = 1 to 5 (stars with planets will have between 1 and 5 planets capable of developing life)
- \(f_{l}\) = 1 (100% of these planets will develop life)
- \(f_{i}\) = 1 (100% of which will develop intelligent life)
- \(f_{c}\) = 0.1 to 0.2 (10–20% of which will be able to communicate)
- \(L\) = 1000 to 100,000,000 communicative civilizations (which will last somewhere between 1000 and 100,000,000 years)

Please use *interval arithmetic* to find the smallest and
largest value of \(N\) under this
model.

## 2. Cost of a tuna can (without the tuna)

If you want to store food in a tin container, you need to make the container. The cost of making a can depends on how much material is required.

How much material is needed to make a tuna can? Please use interval arithmetic to find the interval containing the true value of the area of the can’s surface. The relevant formulas are \[S_1 = π r^2 = \frac{π}{4}D⋅D\\ S_2 = H⋅L = H⋅D⋅π\] where the diameter \(D\) is 85.5mm ± 0.05mm and the height \(H\) is 36.5mm ± 0.05mm.

What are the units of \(S_1\) and \(S_2\)?

## 3. How much tuna can be stored in a tuna can?

Once you have the can, you can put tuna inside. lease use interval arithmetic to find the interval containing the true value of the can’s volume. The relevant formula is \[V=H⋅S_1 = H\cdot \frac{π}{4}D⋅D\]

What are the units of \(V\)?

## 4. If the tuna can grows

We want to store twice as much tuna on the can. How much material do you need?

What is the cheapest way to store tuna?