Using the formula R = (abc)/(4 * Area), we can find the side opposite to angle A. Let a = side opposite to A. Then, a = (4 * Area * R) / (bc) = (4 * 48 * 10) / (b * c).

The maximum area of a triangle with circumradius R is given by the formula Area = (abc)/(4R). For maximum area, the triangle should be equilateral, thus Area = (3√3/4) * (R^2) = (3√3/4) * (5^2) = 25√3/4 cm².

C(n,k) represents the number of ways to choose k items from n, which corresponds to the coefficient of x^k.

For a second-order reaction, if concentration is tripled, the rate increases by 3² = 9 times.

Increasing the concentration of Cu²⁺ increases the cell potential according to the Nernst equation.

According to Le Chatelier's principle, increasing the concentration of reactants will shift the equilibrium position to the right, favoring the formation of products.

Substituting x = 2: 2(2) + 3y = 12 => 4 + 3y = 12 => 3y = 8 => y = 8/3 ≈ 2.67, closest is 3.

Substituting x = 0: 3(0) + 4y = 12 => 4y = 12 => y = 3.

Length AB = √[(4-1)² + (6-2)²] = √[9 + 16] = √25 = 5.

Length CD = |10 - 5| = 5.

Using point-slope form: y - 5 = 1(x - 5) => y = x.

Distance AB = √((3-1)² + (4-2)²) = √(4) = 2.

Distance AB = √((4-1)² + (6-2)²) = √(9 + 16) = √25 = 5.

Calculate distances AB, BC, CA and sum them: AB = 5, BC = 5, CA = 7. Perimeter = 5 + 5 + 7 = 17.

Perimeter = AB + BC + CA = √[(3-1)² + (4-2)²] + √[(5-3)² + (2-4)²] + √[(1-5)² + (2-2)²] = 2.83 + 2.83 + 4 = 10.

Perimeter = AB + BC + CA = √[(4-1)² + (6-2)²] + √[(7-4)² + (2-6)²] + √[(1-7)² + (2-2)²] = 3 + √(9 + 16) + 6 = 3 + 5 + 6 = 14.

Area = 1/2 | x1(y2-y3) + x2(y3-y1) + x3(y1-y2) | = 1/2 | 1(5-2) + 4(2-1) + 7(1-5) | = 1/2 | 3 + 4 - 28 | = 1/2 * 21 = 10.5.

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.

The refractive index n can be calculated using n = 1/sin(θc). For θc = 30°, n = 1/sin(30°) = 1/0.5 = 2.00.

Using the formula n = 1/sin(θc), where θc = 45°, we find n = 1/sin(45°) = 1/0.7071 ≈ 1.41.

Resistance is inversely proportional to cross-sectional area; doubling the area halves the resistance.

Resistance (R) is inversely proportional to cross-sectional area (A). Doubling A halves the resistance.

Using Ohm's Law, R = V / I = 6V / 2A = 3Ω.

Power (P) can be calculated using P = I^2R = (5 A)^2 * 2 Ω = 25 W.

According to Ohm's Law, if the current (I) is doubled and resistance (R) remains constant, the voltage (V) must also double.

Power P = I^2 * R. If I is halved, P becomes (1/2)^2 * R = 1/4 * P, so the power is reduced to a quarter, which means it halves.

Using the current division rule, I1 = I_total * (R2 / (R1 + R2)) = 6A * (4Ω / (2Ω + 4Ω)) = 6A * (4/6) = 4A.

The magnetic field at the center of a circular loop is directly proportional to the current; thus, if the current is doubled, the magnetic field also doubles.