$13$:$53$ C

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$024$

Initially at rest: $d\frac{y}{d}t(x,0)=0$

Use the fourth$-$order Runge$-$Kutta $($RK$4)$ method with time step $h=0\mathrm{.}01s,$ column length $L=2m,El=2000N*m^2,P_0=1000N$ and $=0\mathrm{.}5s^{-1}$ to solve for the lateral displacement $y(L,t)$ at the free end of the column for $t=0$ to $t=5$ seconds. Show all your steps.

$1\mathrm{.}3$ Note that the system can be converted to a system of first$-$order ODEs. Write and solve this system of equations. Report the maximum displacement to four decimal places.

$1\mathrm{.}4$ Write a code that can be used to solve Question $1\mathrm{.}3,$ this code should consider the conversion of the system to first$-$order ODE. You do not need to print the results of the code, but it should be functional. Explain what each command seeks to achieve.

OR

Explain in full how you would solve this on Excel. Your explanations should be detailed and clear.

$(10)$

$[45$ marks$]$

Question $2$

For this question, you can choose to solve everything manually $($by hand$)$ or a computer program, except for Question $2\mathrm{.}2.$ If you solve by coding$/$Excel$,$ you are required to send your code as well for plagiarism checks, this includes the use of GPT tools. Submit your work on myExams and your code to

sibani@unisa.ac$.$za

Population trends across South Africa's metropolitan areas have become critical data points for the country's engineers and urban planners. Consider the contrasting dynamics between Johannesburg's CBD and the rapidly expanding Sandton district. While the traditional city centre has seen fluctuating residential numbers since the post$-$apartheid era, the northern suburbs continue to experience remarkable growth. This poses a challenge for civil engineers as they must design infrastructure that balances population estimates with transport systems

$4$

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and maintaining aging infrastructure, amongst other things. Now, suppose that the expanding Sandton district $($a suburb$)$ population is defined by:

$P_s(t)=\frac{P_{s,\text{m}\text{a}\text{x}}}{1+(\frac{P_{s,\text{m}\text{a}\text{x}}}{P_o}-1)e^{-k_{s}t}}$

and the declining population with time of the CBD $($urban$)$ is defined by:

$P_u(t)=P_{u,\text{m}\text{a}\text{x}}{e}^{-k_{u}t}+P_{u,\text{m}\text{i}\text{n}}$

where $P_{u,\text{m}\text{a}\text{x}},P_{u,\text{m}\text{i}\text{n}},k_u,P_{S,\text{m}\text{a}\text{x}},P_o$ and $k_s$ are empirically derived parameters.

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