Please remember to use complete sentences and explain all of your answers. You may work in groups

of up to 3 people. The final document must contain the names of everyone in the group. You are

expected to turn in a Word document (.doc), as well as a live MATLAB code file (.mlx). Everyone in

the group must turn in the same document. Please, no handwritten parts. For help, you can consult

https://howtotypeanything.com/type-write-math-equations-in-word/ to write math equations

in Word.

1. Introduction

We will continue to use MATLAB in computations. This time, we will be computing buoyancy items. These are typically computed by integrals, but many times the integration is too difficult or impossible to compute directly.

Remember two important rules: You must explain not only what you are doing, but also WHY you are doing it.  The paper must have a narrative structure; do not simply write each question and state an answer.  This means you must use complete sentences.

Use the instructions and formulas to determine a proper way to cut up the problem, how to compute the force involved.

2. Buoyancy

For things partially submerged in water, the water “pushes” up on those objects depending on how deep they are and the area.  The force increases the deeper the object gets.  But the force is diminished if the surface is not perfectly horizontal; there is no buoyancy force on any vertical walls and only partial force on slanted portions. The force pointing up per unit area is given by

F(z,θ) = 9750z cos(θ)

in Newtons per square meter (N/m2 ).  (One kilogram weighs 9.81 Newtons).  Here z is the depth below the water and θ is the angle from horizontal, only between 0 and  .  Adding weight to the boat pushes it down. The buoyancy force pushes it back up to the same amount. In other words, if you want to know how much a boat will go down with a certain of weight, all you have to do is find the buoyant force with that depth.