As should be obvious by now, all your answers should be
**intervals**. They can be written as \(x±\Delta x\).

## 1. Helium balloons

In 2009 Pixar Animation studios released the movie “Up”. On it we see a house floating in the air, pulled up by many helium balloons. What volume of helium is needed to to make a house airborne?

To answer this question, you need to

- Estimate a range for the weight of the house.
- Calculate the weight of a balloon full of air and another full of Helium. You can assume that the balloon has a volume of one cubic meter (\(1m^3\)). This will give you how much weight can be lifted by such balloon.

You can assume that air and helium are ideal gases. The ideal gas law says that \(PV=nRT\) where

- \(P\) is the absolute pressure of the gas,
- \(V\) is the volume of the gas,
- \(n\) is the amount of substance of gas (number of moles),
- \(R\) is the gas constant, equal to \[8.31446261815324 \frac{m^3⋅Pa}{K⋅ mol}\]

## 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

\[ S_1 = π r^2 = \frac{π}{4}D^2\\ S_2 = H⋅L = H⋅D⋅π \]

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 \[V=H⋅S_1 = H\cdot \frac{π}{4}D^2\]

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?