A bulb is constructed using two filaments. A switch allows the...

The power dissipated by a resistance $R$ across a constant voltage $V$ is given by $P = \dfrac{V^2}{R}$.

When two filaments are connected in parallel, the total power is the sum of their individual powers, $P_p = P_1 + P_2$.

Given the possible power outputs are $5 \text{ W}$, $10 \text{ W}$, and $15 \text{ W}$. Since $5 + 10 = 15$, the individual powers of the filaments must be $P_1 = 5 \text{ W}$ and $P_2 = 10 \text{ W}$. The $15 \text{ W}$ output corresponds to the filaments connected in parallel.

For the first filament, the resistance is:

$R_1 = \dfrac{V^2}{P_1} = \dfrac{15^2}{5} = \dfrac{225}{5} = 45 \Omega$

For the second filament, the resistance is:

$R_2 = \dfrac{V^2}{P_2} = \dfrac{15^2}{10} = \dfrac{225}{10} = 22.5 \Omega$

The resistances of the filaments are $45 \Omega$ and $22.5 \Omega$.