Question

what is the equation of the x-coordinate of the vertex of the daily vs total cases parabola?Question 6 and 7

Answer 1

Answer:

$N=\frac{N_{\infty}}{2}$

Vertex: (150, 17.33)

Explanation:

The equation for the parabola is given as:

$\begin{aligned} & K=-\frac{r}{N_{\infty}}N^2+rN\text{ where:} \\ & K=\text{ Daily New Cases, } \\ & N=\text{ Total Cumulative Cases (at a particular time }t\text{ ) } \\ & N_{\infty}=\text{ maximum possible total cases } \\ & r\text{ is the exponential growth rate of the pandemic if }N=N_0e^{rt}\end{aligned}$

Part 6

The x-coordinate of the vertex of the parabola is the equation of the axis of symmetry.

We can find the equation of the axis of symmetry using the formula:

$x=-\frac{b}{2a}$

From the equation for K:

$\begin{gathered} K=-\frac{r}{N_{\infty}}N^2+rN\implies a=-\frac{r}{N_{\infty}},b=r \\ \implies x=-r\div(-\frac{2r}{N_{\infty}}) \\ =r\times\frac{N_{\infty}}{2r} \\ N=\frac{N_{\infty}}{2} \end{gathered}$

The equation of the x-coordinate of the vertex of the daily vs total cases parabola is:

$N=\frac{N_{\infty}}{2}$

Part 7

• From part (4), the growth rate, r= ln(2)/3.

,

• Given that N∞ = 300 million

The coordinates of the vertex will be:

$(N,K)=(\frac{N_{\infty}}{2},-\frac{r}{N_{\infty}}N^2+rN)$

Replace N in the y-coordinate with the equation obtained from part(6).

$\begin{gathered} (N,K)=(\frac{N_{\infty}}{2},-\frac{r}{N_{\infty}}\times(\frac{N_{\infty}}{2})^2+\frac{rN_{\infty}}{2}) \\ =(\frac{N_{\infty}}{2},-\frac{rN_{\infty}}{4}+\frac{rN_{\infty}}{2}) \\ =(\frac{N_{\infty}}{2},\frac{2rN_{\infty}-rN_{\infty}}{4}) \\ =(\frac{N_{\infty}}{2},\frac{rN_{\infty}}{4}) \end{gathered}$

Substitute the given values:

$(N,K)=(\frac{300}{2},\frac{\frac{\ln(2)}{3}\times300}{4})=(150,25\ln (2))$

The coordinates of the vertex will be (150, 17.33).