Opportunities for Mathematical Application

Running head: MATH3071
MATH307
[Name]
[Institutional Affiliation]

MATH3072
Abstract
In businesses and many industries, there have been computers that have been
used, and are being used even now, to solve problems mathematically and activities. In the
last 25 years, the opportunities for mathematical application have been on the increase as
more cases that requires the mathematical input increases as well. Those applying
mathematics must realise that the application of mathematics is a totally different activity
from the learning of mathematics. The skills that one needs to apply mathematics are very
different from the ones that are needed for understanding of mathematical concepts, which at
the end helps in proving the theories or help solve the equations (Dilwyn & Hamson,
2001). Modelling is the art of dealing with real problems, and the problems are dealt with
mathematically. The good thing with modelling is that it involves discussion so that there’s
clarity of the problems and the variables are identified, estimated, and approximated. This
paper deals with modelling of a public facility occupancy. Where a building’s occupancy at
any given period is known.
Introduction
Members of the public from every walk of life find access to every public
facility, for various reasons, healthcare, education, social gatherings, and department of
corrections. That is why whenever the name “public” is mentioned, then there must be
planning involved. Part of this planning is the occupancy of the facility, where the number of
people occupying that facility is known. The reason this is so, is for emergency incidences
such as fire, shootout or any emergency that might require evacuation. The reasoning behind
planning also is for averting overcrowding. Overcrowding in learning facilities for example,
gives teachers headache since there’s the teacher to student ratio which indicates the resource
sharing capabilities, also this is important to avoid making teachers being fatigued by
teaching (OECD , 2019). Overcrowding also makes spreading of diseases easy, like flu or
common cold or coughing. When people are spaced out properly, then the disease spreading
is not as fast. Public safety is a promise that every government must give and assure its
citizen.
In this modelling problem, we are dealing or want to help the evacuators and the
evacuees. Since this has been a challenge to many governments or authorities that evacuates
people. the authorities that deals with evacuators, can work with the authorities that approve
the construction drawings so that certain things are not left out while building. This will make

MATH3073
sure that the exit routes are strategically constructed for people to leave through them. After
commissioning, each building will have an evacuation procedure of how to leave the
building. In matters evacuation, time is of essence, they would want to know what time it will
take for a number of people to leave a building or a room. The people leaving the room must
be safe while leaving the room so that there is no stampede when leaving in a hurry. So, that
is why modelling is necessary to avert such issues and others. So that there’s a specific
number of students when a room is full to capacity.
Introduction to differential equations
Tasks
Many public facilities have signs in rooms used for public gatherings which state that it is
“unlawful” for the rooms to be occupied by more than a specified number of people.
Presumably, this number is based on the speed with which people in the room could be
evacuated from the room’s exits in case of an emergency. Similarly, elevators and other
facilities often have “maximum capacities” posted.
i. Develop a mathematical model for deciding what number to post on such a sign as
being the “lawful capacity”.
ii. As part of your solution discuss criteria, other than public safety in the case of a
fire or other emergency, that might govern the number of people considered
“unlawful” to occupy the room (or space).
iii. Also, for the model that you construct, consider the differences between a room
with movable furniture such as the TLC (with tables and chairs), the Y, a
classroom in Joy, and a biology/computer lab with a pattern of rows and aisles.
iv. You may wish to compare and contrast what might be done for a variety of
different environments: elevator, classroom, or the gym. Gatherings such as rock
concerts and soccer tournaments may present special conditions
v. Apply your model in one or more public facilities at UWT campus and compare
your results with the stated capacity if one is posted
Solutions
From the start, we might be required to deal with knowing how large the room is, the number of exit
stairs there are etc. To model this, we will take some assumptions
Let the building be a school or a college in one floor