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=(N_(\infty))/(2)`

Vertex: (150, 17.33)

Explanation:

The equation for the parabola is given as:

`\begin{aligned} & K=-(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=-(b)/(2a)`

From the equation for K:

`\begin{gathered} K=-(r)/(N_(\infty))N^2+rN\implies a=-(r)/(N_(\infty)),b=r \\ \implies x=-r/(-(2r)/(N_(\infty))) \\ =r*(N_(\infty))/(2r) \\ N=(N_(\infty))/(2) \end{gathered}`

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

`N=(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)=((N_(\infty))/(2),-(r)/(N_(\infty))N^2+rN)`

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

`\begin{gathered} (N,K)=((N_(\infty))/(2),-(r)/(N_(\infty))*((N_(\infty))/(2))^2+(rN_(\infty))/(2)) \\ =((N_(\infty))/(2),-(rN_(\infty))/(4)+(rN_(\infty))/(2)) \\ =((N_(\infty))/(2),(2rN_(\infty)-rN_(\infty))/(4)) \\ =((N_(\infty))/(2),(rN_(\infty))/(4)) \end{gathered}`

Substitute the given values:

`(N,K)=((300)/(2),((\ln(2))/(3)*300)/(4))=(150,25\ln (2))`

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