## Task 32/66

An electric heater contains two resistors $R_{1}$ and $R_{2}$, which can be switched in four stages $A, B, C$, and $D$.

In stage $A$, $R_{1}$ and $R_{2}$ are connected in parallel, in stages $B$ and $C$ one of the two resistors is in operation, and in stage $D$ $R_{1}$ and $R_{2}$ are connected in series.

The resistors should be dimensioned such that at a voltage of $22 \mathrm{~V}$ a maximum power of $6000 \mathrm{~W}$ occurs and the powers in the different stages form a geometric sequence. How large must the resistors be and what powers occur in the individual stages?
It is $N=U \cdot I=\frac{U^{2}}{R}$, where $N$ denotes power, $U$ voltage, $I$ current, and $R$ resistance. From this, the maximum power follows as

$$
N_{\max }=U^{2} \cdot \frac{1}{R_{\min }} \quad \text { or } \quad R_{\min }=\frac{U^{2}}{N_{\max }}=\frac{220^{2}}{6000} \Omega
$$

According to Kirchhoff's law of current division, $R_{\text {min }}$ for the parallel connection of $R_{1}$ and $R_{2}$ is

$$
R_{\min }=\frac{R_{1} R_{2}}{R_{1}+R_{2}} \quad \text { thus } \quad R_{\min }=\frac{R_{1} R_{2}}{R_{1}+R_{2}}=\frac{220^{2}}{6000} \Omega
$$

Let $N_{\max }=N_{A}=a N_{B}=a^{2} N_{C}=a^{3} N_{D}$, where $a$ is the quotient of the required geometric sequence and the index in $N$ indicates the switching stage. Then,

$$
N_{B}=\frac{U^{2}}{R_{1}} ; \quad N_{C}=\frac{U^{2}}{R_{2}} ; \quad N_{D}=\frac{U^{2}}{R_{1}+R_{2}} \quad \text { or } \quad R_{1}=\frac{U^{2}}{N_{B}}=\frac{a U^{2}}{N_{A}} ; \quad R_{2}=\frac{U^{2}}{N_{C}}=\frac{a U^{2}}{N_{D}}
$$

and from the equation for $R_{\text {min }}$, by substitution and simplification, $a^{2}-a-1=0$ or $a=0.5(1+\sqrt{5})$. From the equations, the resistances $R_{1}$ and $R_{2}$ can now be calculated:

$$
R_{1}=\frac{220^{2} \cdot 0.5(1+\sqrt{5})}{6000} \approx 13 \Omega \quad, \quad R_{2}=\frac{220^{2} \cdot 0.5^{2}(1+\sqrt{5})^{2}}{6000} \approx 21.1 \Omega
$$

For the powers, $N_{B} \approx 3710 \mathrm{~W}, N_{C} \approx 2290 \mathrm{~W}, N_{D} \approx 1420 \mathrm{~W}$. It is shown that the power stages have an additional relationship: $N_{B}=N_{C}+N_{D}$ and $N_{A}=N_{B}+N_{C}$.