A platinum resistance thermometer is constructed which reads `,0^0` at ice point and `100^0` at steam point. Let `t_p` denote temperature on this scale and let t denote the temperature on a mercury on a mercury thermometer scale. The resistance of the platinum coil varies with `t` as `R_1 = R_0 (1+alphat + betat^2)`. Derive an expression for the resistance as a function of `T_p`.

Explanation

To derive the expression for the resistance of a platinum resistance thermometer as a function of the temperature \( T_p \) on its scale, we start with the given information and relationships.

### Step 1: Define the relationship between \( T_p \) and resistance \( R_{TP} \)

We know that the temperature \( T_p \) on the platinum thermometer scale can be defined in terms of its resistance \( R_{TP} \) as follows:

\[
T_p = \frac{R_{TP} - R_0}{R_{100} - R_0} \times 100
...