Equation of resistance from given graph

[Aurelius120]

Homework Statement

Given the graph of $lnR$ vs $T^{-2}$, predict the relationship between resistance and temperature

Relevant Equations

NA

From the graph:
$$lnR(T)=\frac{-lnR(0)T^2_○}{T^2}+lnR(0)$$
I have assumed $R(0)$ to be the value of $R$ at $1/T^2=0$ and $T_○$ to be the value of $T$ at $lnR(T)=0$
From this I get,
$$R(T)=e^{lnR(0)×\left(1-\frac{T_○^2}{T^2}\right)}$$
$$R(T)=R(0)^{\left(1-\frac{T_○^2}{T^2}\right)}$$
This does not match with any of the options.
I couldn't reduce it any of the options either.

Maybe my choice of $R(0)$ and the one in the options refer to different quantities. Even so I could not get to the right answer(given option-c in the book).

Please help

 

[haruspex]

I have assumed $R(0)$ to be the value of $R$ at $1/T^2=0$ and $T_○$ to be the value of $T$ at $lnR(T)=0$

I would assume R₀ is the value of R when T=T₀.

 

[Orodruin]

It is unclear to me why you have included $\ln R_0$ in the linear term. There is no reason to.

I suggest you write the linear equation on a more agnostic form $\ln R = -k/T^2 + m$, solve for $R$, and only then try to identify $R_0$ and $T_0$.

 

[Orodruin]

I would assume R₀ is the value of R when T=T₀.

I would not assume anything. I would wait with introducing the constants $R_0$ and $T_0$ until I can introduce them to get the result on one of the given forms. As should be clear from the given forms, $R_0$ is the value of $R$ in particular limits depending on the option. ($T = 0$ for a and b, $T \to \infty$ for c, and $T = 1$ for d)

 

[Aurelius120]

It is unclear to me why you have included $\ln R_0$ in the linear term. There is no reason to.

I suggest you write the linear equation on a more agnostic form $\ln R = -k/T^2 + m$, solve for $R$, and only then try to identify $R_0$ and $T_0$.

Done
$$ln(R(T))=\frac{-k}{T^2}+m$$
$$R(T)=e^{(-k/T^2+m)}=e^m×e^{-k/T^2}=R_○e^{-T_○^2/T^2}$$