Using the balance condition R1/R2 = R3/R4, we find R4 = (R2 * R3) / R1 = (20 * 15) / 10 = 30Ω.

Using the balance condition R1/R2 = R3/R4, we have 20/30 = 10/x, solving gives x = 20Ω.

The condition for balance in a Wheatstone bridge is R1/R2 = R3/R4.

Resistance R = ρ(L/A) = (1.5 x 10^-8)(3)/(0.5 x 10^-6) = 0.09 Ω.

Resistance R = ρ * (L / A) = 2 x 10^-8 * (3 / 0.5 x 10^-6) = 0.12 Ω.

Resistance R is directly proportional to resistivity; if resistivity doubles, resistance also doubles.

Resistance R is directly proportional to resistivity ρ, so if ρ is doubled, R also doubles.

Resistance is directly proportional to resistivity; halving resistivity halves the resistance.

Resistance is directly proportional to resistivity; halving resistivity halves the resistance.

Resistance is directly proportional to resistivity; halving resistivity halves the resistance.

A superconductor has zero resistivity, which means it has zero resistance.

Resistance R = ρ * (L / A) = 1.68 x 10^-8 * (100 / (π * (0.5 x 10^-3)²)) = 0.168 Ω.

Using v_rms = sqrt(3RT/M), we can rearrange to find T = (v_rms^2 * M) / (3R) = 300 K.

Using the formula v_rms = sqrt((3RT)/M), we can rearrange to find T = (v_rms^2 * M)/(3R). Plugging in the values gives T = 600 K.

The RMS speed is proportional to the square root of the temperature. Therefore, at 600 K, the RMS speed will be 300 * sqrt(2) m/s.

The RMS speed is proportional to the square root of the temperature. Therefore, at 200 K, the RMS speed will be 300 * sqrt(200/400) = 150 m/s.

RMS speed is proportional to the square root of temperature. v_rms at 800 K = 300 * sqrt(800/400) = 300 * sqrt(2) = 600 m/s.

The kinetic energy per molecule is given by KE = 0.5 * m * v^2. Substituting v = 300 m/s gives the correct expression.

The RMS speed is proportional to the square root of the temperature. If the temperature is doubled, the RMS speed increases by a factor of sqrt(2). Therefore, the new RMS speed will be 300 * sqrt(2), which is approximately 600 m/s.

Using the formula v_rms = sqrt((3RT)/M), we can rearrange to find T: T = (M * v_rms^2) / (3R). Substituting M = 0.016 kg/mol and v_rms = 400 m/s gives T = 400 K.

The RMS speed increases with the square root of the temperature. Therefore, at 600 K, the RMS speed will be 400 * sqrt(2), which is approximately 800 m/s.

Kinetic energy per molecule = (1/2)mv^2. Using v = 400 m/s and m = M/N_A, we find KE = 0.5 mJ.

The average speed is related to RMS speed by the relation v_avg = (2/3) * v_rms. Thus, v_avg = (2/3) * 400 = 267 m/s.

The average speed of gas molecules is approximately 0.8 times the RMS speed, so average speed = 0.8 * 400 m/s = 320 m/s.

The speed at 1/2 of the RMS speed is simply 500 m/s / 2 = 250 m/s.

The average speed of gas molecules is related to the RMS speed by the relation v_avg = (v_rms * sqrt(8/3)). Therefore, the average speed is approximately 400 m/s.

The RMS speed is an average measure; individual molecular speeds will vary around this value.

Using v_rms = sqrt(3RT/M), we rearrange to find T = (v_rms^2 * M) / (3R). Plugging in values gives T approximately 300 K.

Using Vieta's formulas, a = 1 since the product of the roots (3 * -2) = -6 and sum (3 + -2) = 1.

Using the sum of roots (-b/a = 3 + (-2) = 1), we find b = -1.