Letter A as shown in Fig. 4.57 has resistance on each side of arm. Calculate the total resistance between two ends of the legs.

Answer

To calculate the total resistance between the two ends of the legs of the letter A, we will follow these steps:

### Step 1: Identify the resistances in series
The letter A has two resistances on the upper part of the letter: one is 10 ohms and the other is 5 ohms. Since these two resistances are in series, we can calculate the equivalent resistance (R1) of this part.

**Calculation:**
\[ R_1 = R_{10} + R_{5} = 10 \, \Omega + 5 \, \Omega = 15 \, \Omega \]

### Step 2: Identify the resistances in parallel
Now, we have the equivalent resistance of the upper part (15 ohms) and the resistance on the other leg (5 ohms). These two resistances (15 ohms and 5 ohms) are in parallel.

**Calculation:**
Using the formula for resistances in parallel:
\[ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} \]
Where \( R_1 = 15 \, \Omega \) and \( R_2 = 5 \, \Omega \):
\[ \frac{1}{R_{eq}} = \frac{1}{15} + \frac{1}{5} \]
Finding a common denominator (15):
\[ \frac{1}{R_{eq}} = \frac{1}{15} + \frac{3}{15} = \frac{4}{15} \]
Thus,
\[ R_{eq} = \frac{15}{4} \, \Omega = 3.75 \, \Omega \]

### Step 3: Combine the resistances in series
Next, we need to add the resistance of the lower part of the letter A, which is 10 ohms, to the equivalent resistance we just calculated (3.75 ohms).

**Calculation:**
\[ R_{total} = R_{lower} + R_{eq} = 10 \, \Omega + \frac{15}{4} \, \Omega \]
Converting 10 ohms to a fraction:
\[ R_{total} = \frac{40}{4} + \frac{15}{4} = \frac{55}{4} \, \Omega = 13.75 \, \Omega \]

### Final Answer
The total resistance between the two ends of the legs of the letter A is:
\[ R_{total} = \frac{55}{4} \, \Omega = 13.75 \, \Omega \]

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