The minimum cost occurs at x = -b/(2a) = -12/(2*3) = -2. C(-2) = 3(-2)^2 + 12(-2) + 5 = 8.

The minimum cost occurs at x = -b/(2a) = -12/(2*3) = -2. C(-2) = 3(-2)^2 + 12(-2) + 5 = 8.

The minimum cost occurs at x = -b/(2a) = -20/(2*5) = -2. C(-2) = 5(-2)^2 + 20(-2) + 100 = 120.

RMS current I_rms = I0/√2 for a cosine function.

The RMS current is I_rms = I_peak / √2 = 5√2 / √2 = 5 A.

Power is proportional to the square of the current (P = I²R). If I is tripled, P increases by 3² = 9 times.

Circumference = πd; C = π * 20 ≈ 62.83 cm.

F = E * q = 1000 N/C * 5 × 10^-6 C = 0.005 N.

The nth term of a geometric series is given by a_n = ar^(n-1). Here, a = 4, r = 2, n = 7. So, a_7 = 4 * 2^(7-1) = 4 * 64 = 256.

In an arithmetic series, the last term can be expressed as a + (n-1)d. Here, 48 = 12 + (10-1)d. Thus, 48 - 12 = 9d, giving d = 4.

The time period T is the reciprocal of frequency f, T = 1/f. Therefore, T = 1/440 ≈ 0.00227 s.

Wavelength (λ) is inversely proportional to frequency (f) when speed (v) is constant: λ = v/f. If f is doubled, λ is halved.

Wavelength is inversely proportional to frequency. If frequency is halved, wavelength doubles.

Xl = 2πfL; if f is doubled, Xl also doubles.

Inductive reactance XL = 2πfL, so if frequency f is doubled, XL halves.

The capacitive reactance Xc = 1/(2πfC) halves when the frequency f is doubled.

F is inversely proportional to r². If r is doubled, F becomes F/4 = 20 N / 4 = 5 N.

Xl = 2πfL = 2π × 60 × 0.5 = 31.42 ohms.

Total impedance Z = √(R² + (XL)²) = √(10² + 30²) = √(100 + 900) = √1000 = 31.62Ω.

Heat required = mass × latent heat = 50 g × 334 J/g = 16700 J.

Energy required = mass * latent heat = 0.5 kg * 2260 kJ/kg = 1130 kJ.

Setting y = 0 gives 2x = 6, thus x = 3. The point of intersection is (3, 0).

Setting y = 0 gives 2x = 6, thus x = 3. The x-coordinate of the intersection point is 3.

Setting x = 0 gives 4y = 12, thus y = 3. The point of intersection is (0, 3).

Setting y = 0 in the equation gives 3x + 12 = 0, thus x = -4.

Setting x = 0 gives -4y + 12 = 0, thus y = -3. The point of intersection is (0, -3).

Rearranging gives y = (3/4)x + 3. The slope of the line is 3/4, so the slope of the parallel line is -3/4.

Parallel lines have the same slope. The slope of the given line is 3/4, which is the same as the slope of 6x - 8y + 24 = 0.

Rearranging to y = -7/2x + 7 shows that the slope is -7/2.

The force experienced by a current-carrying conductor in a magnetic field is directly proportional to the magnetic field strength. Therefore, if the magnetic field strength is doubled, the force also doubles.