To find the resistance of the two coils, let's denote the resistances of the two coils as \( R_1 \) and \( R_2 \).
### Step 1: Set up equations for series and parallel combinations
When the resistors are connected in series, the equivalent resistance \( R_s \) is given by:
\[
R_s = R_1 + R_2
\]
The maximum resistance obtained is 16 ohms, so we have:
\[
R_1 + R_2 = 16 \quad \text{(Equation 1)}
\]
When the resistors are connected in parallel, the equivalent resistance \( R_p \) is given by:
\[
R_p = \frac{R_1 R_2}{R_1 + R_2}
\]
The minimum resistance obtained is 3 ohms, so we have:
\[
\frac{R_1 R_2}{R_1 + R_2} = 3 \quad \text{(Equation 2)}
\]
### Step 2: Substitute Equation 1 into Equation 2
From Equation 1, we can express \( R_1 + R_2 \) as 16. Substituting this into Equation 2 gives:
\[
\frac{R_1 R_2}{16} = 3
\]
Multiplying both sides by 16:
\[
R_1 R_2 = 48 \quad \text{(Equation 3)}
\]
### Step 3: Solve the system of equations
Now we have two equations:
1. \( R_1 + R_2 = 16 \)
2. \( R_1 R_2 = 48 \)
We can use these equations to find \( R_1 \) and \( R_2 \). From Equation 1, we can express \( R_2 \) in terms of \( R_1 \):
\[
R_2 = 16 - R_1
\]
Substituting this into Equation 3:
\[
R_1(16 - R_1) = 48
\]
Expanding this gives:
\[
16R_1 - R_1^2 = 48
\]
Rearranging gives us a quadratic equation:
\[
R_1^2 - 16R_1 + 48 = 0
\]
### Step 4: Solve the quadratic equation
To solve the quadratic equation \( R_1^2 - 16R_1 + 48 = 0 \), we can use the quadratic formula:
\[
R_1 = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here, \( a = 1 \), \( b = -16 \), and \( c = 48 \):
\[
R_1 = \frac{16 \pm \sqrt{(-16)^2 - 4 \cdot 1 \cdot 48}}{2 \cdot 1}
\]
Calculating the discriminant:
\[
R_1 = \frac{16 \pm \sqrt{256 - 192}}{2}
\]
\[
R_1 = \frac{16 \pm \sqrt{64}}{2}
\]
\[
R_1 = \frac{16 \pm 8}{2}
\]
This gives us two possible values for \( R_1 \):
\[
R_1 = \frac{24}{2} = 12 \quad \text{or} \quad R_1 = \frac{8}{2} = 4
\]
### Step 5: Find \( R_2 \)
Using \( R_1 + R_2 = 16 \):
- If \( R_1 = 12 \), then \( R_2 = 16 - 12 = 4 \).
- If \( R_1 = 4 \), then \( R_2 = 16 - 4 = 12 \).
Thus, the resistances of the two coils are:
\[
R_1 = 12 \, \text{ohms}, \quad R_2 = 4 \, \text{ohms}
\]
### Final Answer
The resistance of the two coils are \( 12 \, \text{ohms} \) and \( 4 \, \text{ohms} \).
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