CPSC120B - Activity 11

Equilateral Area

Create a Python program that defines the function equilateral_area(edge_length) that computes the area of an equilateral triangle. The parameter edge_length is a float that specifies the length of the triangle's edges. The function should return the area of the triangle as a float. The area of an equilateral triangle and be computed with the equation:

$a = \frac{\sqrt{3}}{4} \times l ^ 2$

Where $a$ is the area of the triangle and $l$ is the length of an edge of the triangle.

Test Cases

Create your own! Use this template:

print('Input:', ?, '\tActual:', equilateral_area(?), '\tExpected:', ?)

Sierpinski

Create a Python program that defines the function sierpinski_area(edge_length) that computes the area of a Sierpinski triangle of order one. A Sierpinski Triangle of order one is an equilateral triangle with a triangular hole. The vertices of the hole triangle are the mid points of the Sierpinski triangle's edges. For example:

Copy and paste your solution to the previous exercise into this one and use it to write the sierpinski_area function.

Test Cases

Create your own! Use this template:

print('Input:', ?, '\tActual:', sierpinski_area(?), '\tExpected:', ?)

Trees

When working with functions, abstraction is key. You will soon find yourself in a position where you start thinking about problems in functions. Being able to break a complex problem, like an advanced drawing, into smaller and simpler problems which can be solved with a function can make your code a lot easier to write, and a lot easier to read.

Details

Create a Python program that defines the function draw_tree(x, y) that uses the turtle module to draw a tree. The x and y location specified is the center of the base of the triangle making up the tree. The tree should look like the tree in the example below.

Notice that a tree is defined by a brown square, and a filled green triangle. To simplify the draw_tree function, also define the function draw_triangle(x, y), which draws a filled, green isosceles triangle. The triangle can be any height, but the length of the base should be the same as the height.

Example

Drawing the trunk of the tree is simple, just draw a filled square.
Drawing the tree's canopy is easier if you use the turtle.goto function. To help figure out the coordinates of the triangle corners, make a drawing of the triangle inside of square. The triangle fits inside of a square since its width is the same as its height.

Challenge

Modify your program so that you can specify the size of the tree. We will define the size of the tree to be the witdh of the tree (and the height of the triangle). Use this new function to draw yourself a forest of trees.

Twin Paradox

In the year 2047, a set of twins were born. One of these twins was placed in a rocket and launched towards a distant star at a significant percentage of the speed of light. Unlike Superman, however, the rocket was programmed to return home after reaching its destination. Because of the laws of relativity, the twin who was launched into space at close to the speed of light will have aged less than their sibling. The question is, how much younger is the spacefaring twin?

Details

Create a Python program that defines the function compute_age_difference(distance_lightyears, percentage_of_light) that uses the laws of relativity to compute how much younger the space traveling twin is. The parameter distance_lightyears is a float representing a distance traveled, in light-years. The parameter percentage_of_light is a float in the range $(0, 100)$ that represents the percentage of the speed of light that the twin traveled. The function should return how many years younger the twin is.

The function should assume that all acceleration is instant, which simplifies the calculations. The speed of light is $2.99 \times 10 ^ 8 m / s^2$ and the number of seconds passed on earth can be computed using equation:

$time\_on\_earth = 2 \frac{distance\_meters}{speed\_traveled}$

The fraction of time passed on the rocket as compared to Earth can be computed using the dilation equation:

$dilation = \sqrt{1 - \frac{speed\_traveled ^ 2}{speed\_of\_light ^ 2}}$

Test Cases

Create your own! Use this template:

print('Input:', ?, ',', ?, '\tActual:', compute_age_difference(?, ?), '\tExpected: ?')

Write a function called lightyears_to_meters(lightyears), which converts a distance in light years to meters. One light year is $9.56 \times 10 ^ {15}$ meters.
The time equation listed above returns the time that has passed on Earth in seconds. Write a function called seconds_to_years(seconds), which returns a floating point value representing how many years that number of seconds represents.
Using the above equations, you simply need to subtract the time that passed on the ship from the time passed on earth.

Challenge

What if, instead of just launching one of the children into space, these crazy scientists launched both? Likely, they would have tried to send two of the siblings off into space, but at different speeds. Write a new function called compute_relativistic_age_difference, which computes the difference between two siblings who were launched into space at differing speeds.