In this problem, we calculate what voltage is needed to produce a 0.62 A current in the 250 $Omega$ resistor. To solve the problem we remember $textbf{Ohm’s law}$:

$$
R=frac{V}{I}
$$

Expressing the voltage from this we have:

$$
V=R I
$$

Putting in the information from the problem statement we have:

$$
V = 250 mathrm{Omega} cdot 0.62 mathrm{~A}
$$

$$
boxed{color{#c34632}V= 155 mathrm{~V}}
$$

$$
V= 155 mathrm{~V}
$$

Here we calculate the resistance of the wire if we know that when the voltage of 18 V is applied, the wire conducts 0.35 A of current. To solve this we use Ohm’s law:

$$
R=frac{V}{I} = frac{18 mathrm{~V}}{0.35 mathrm{~A}}
$$

We have the result of:

$$
boxed{color{#c34632}R=51.43 mathrm{Omega}}
$$

$$
R=51.43 mathrm{Omega}
$$