Skip to main content

Jungle Speed vs Solids with Known Cross Sections

As a kid, I played a game called Jungle Speed. Here's a picture for reference, but essentially everyone playing flipped one card over at a time and when a specific pattern appeared, your goal was to grab the yellow stick before anyone else could. Well, when I thought about calculating the volume of a solid with known cross sections, the first thing that came to mind was that yellow stick. I know that sounds really random, but that's what I decided to try and recreate.

The cross sections of the yellow stick are circles, but in order to better explain cross sections I chose to use rectangular prisms as my cross sections as they are easy to visualize and have a basic volume equation. I now had to think about what equations could make up the wave like sides of the yellow stick and I figured a trig function would be best suited for a wave like shape. I chose the following functions and evaluated the volume of known cross sections of squares between \(x=0\) and \(x=10\):

\[f(x) = cos(x) + 2\] and \[g(x) = sin(x)\]

In order to approximate the volume of my solid with square cross sections, I had to calculate the volume of each of my ten rectangular prisms and add them together. Since the volume of a rectangular prism with a square base is \(V=b^2*h\), I calculated the difference between \(f(x)-g(x)\) to be my base and used a height of \(1\) for each prism to ensure an even spread. I used the midpoint of each cross section to get the most acurrate volume of my object. Below is a table showing a summary of my calculations and the total approximate volume of \(39.71units^3\):



In order to calculate the actual volume of my solid, I had to calculate the integral of the area of each cross section with respect to \(x\). The integral below shows the actual volume of my solid is about \(40.17units^3\):

\begin{aligned}\int _{0}^{10}\left[ \left( \cos x+2\right) -\sin x\right] ^{2}dx\\ .\end{aligned} \begin{aligned}=\int _{0}^{10}\cos ^{2}\left( x\right) +400s\left( x\right) -2sin\left( x\right) \cos \left( x\right) -4\sin \left( x\right) +sin^{2}\left( x\right) +4dx\\ =-\sin ^{2}\left( 10\right) +4\sin \left( 10\right) +4\cos \left( 10\right) +46=40.17units^{3}\end{aligned}

The solid that I ended up creating looked a little less like the yellow stick from my game and a little more like when the kid you babysit doesn't understand physics and tries to make a tower with their blocks. But, you can definitely still see the wave like sine and cosine functions that I wanted to see, so it wasn't a total fail. The image below shows the solid I will attempt to print. Once I created my solid, I scaled it down to 40% to create the dimensions \(101.7mm\) x \(40.28mm\) x \(34.69mm\).
Labels:

Comments

Post a Comment

Popular posts from this blog

Do Over: Integration Over a Region in a Plane

Throughout the semester we have covered a variety of topics and how their mathematical orientation applies to real world scenarios. One topic we discussed, and I would like to revisit, is integration over a region in a plane which involves calculating a double integral. Integrating functions of two variables allows us to calculate the volume under the function in a 3D space. You can see a more in depth description and my previous example in my blog post, https://ukyma391.blogspot.com/2021/09/integration-for-over-regions-in-plane_27.html . I want to revisit this topic because in my previous attempt my volume calculations were incorrect, and my print lacked structural stability. I believed this print and calculation was the topic I could most improve on and wanted to give it another chance. What needed Improvement? The function used previously was f(x) = cos(xy) bounded on [-3,3] x [-1,3]. After solving for the estimated and actual volume, it was difficult to represent in a print...
Post a Comment
Read more

Minimal Surfaces

Minimum surfaces can be described in many equivalent ways. Today, we are going to focus on minimum surfaces by defining it using curvature. A surface is a minimum surface if and only if the mean curvature at every point is zero. This means that every point on the surface is a saddle point with equal and opposite curvature allowing the smallest surface area possible to form. Curvature helps define a minimal surface by looking at the normal vector. For a surface in R 3 , there is a tangent plane at each point. At each point in the surface, there is a normal vector perpendicular to the tangent plane. Then, we can intersect any plane that contains the normal vector with the surface to get a curve. Therefore, the mean curvature of a surface is defined by the following equation. Where theta is an angle from a starting plane that contains the normal vector. For this week’s project, we will be demonstrating minimum surfaces with a frame and soap bubbles! How It Works Minimum surfac...
Post a Comment
Read more

Ruled Surfaces : Trefoil

Ruled Surfaces : Trefoil A ruled surface is a surface that consists straight lines, called rulings, which lie upon the surface. These surfaces are formed of a set of points that are "swept" by a straight line. This is relatively intuitive once you see a good visual, but can be a bit abstract without that concrete example. A very basic example of a ruled surface is a cylinder; if we have a straight line and move it in a circle we create a cylinder made entirely of straight line. Note that the surface will only be a cylinder if all the lines are parallel. If the lines are not parallel we can create hyperboloids and cones depending on how much we have rotated. The rotation we are describing here is not a simple turning action, but more of a twisting motion—less like rotating a can by turning it and more like wringing out a washcloth by twisting it. Specifically, a cylinder is essentially two circles connected by rulings, if we keep one of the circles...
Post a Comment
Read more