The drop of oil spreads on water because the surface tension of water is higher than that of oil, causing the oil to spread out.

A drop of water is spherical due to surface tension, which minimizes the surface area.

Daily production = 2000 items / 5 days = 400 items/day. In 12 days, it produces 400 × 12 = 4800 items.

The possible combinations of children are BB, BG, GB, GG. Given that at least one is a boy, we can eliminate GG, leaving us with BB, BG, GB. Out of these 3 combinations, only 1 is BB. Therefore, the probability is 1/3.

The possible combinations are BB, BG, GB. Given at least one is a boy, the combinations are BB, BG, GB. The probability of both being boys is 1/3.

The possible combinations of children are BB, BG, GB, GG. Out of these, only 1 combination is both boys (BB). Thus, the probability is 1/4.

The only combinations with at least one boy are: BBB, BBG, BGB, GBB, BGG, GBG, GGB. Out of these, all combinations except BBB have at least one girl. Thus, P(At least one girl | At least one boy) = 6/7.

The only scenario where there are no girls is if all are boys. The probability of at least one girl given at least one boy is 3/4.

Total income = Average income × Number of members = 5000 × 4 = $20000.

Current total age = 5 + 10 + 15 = 30. To maintain an average of 10 with 4 children, total age must be 40. Therefore, the new child's age must be 40 - 30 = 10.

For a fixed area, the minimum perimeter occurs when the rectangle is a square. Thus, dimensions are approximately 14 m by 14.28 m.

For total internal reflection to occur in a fiber optic cable, the core must have a higher refractive index than the cladding.

Critical angle θc = sin⁻¹(n2/n1) = sin⁻¹(1.5/1.6) ≈ 38.7°.

Critical angle θc = sin⁻¹(n2/n1) = sin⁻¹(1.4/1.5) ≈ 42.0°.

For total internal reflection, the core must have a higher refractive index than the cladding, so it must be greater than 1.45.

The cladding has a lower refractive index than the core, ensuring that light remains within the core by total internal reflection.

The cladding has a lower refractive index than the core, ensuring that light is kept within the core through total internal reflection.

Angular momentum is conserved; it remains constant, but her angular velocity increases.

By conservation of angular momentum, if the moment of inertia decreases, the angular velocity must increase.

Angular momentum remains the same; however, her angular velocity increases due to a decrease in moment of inertia.

Pulling her arms in decreases her moment of inertia, causing her rotational speed to increase to conserve angular momentum.

By conservation of angular momentum, pulling arms in decreases moment of inertia, thus increasing angular velocity.

The correct form is 'A flock of birds was flying over the lake.' The subject 'flock' is singular, so the verb should be 'was'.

Using Poiseuille's law, the flow rate Q = (π * r^4 * ΔP) / (8 * η * L). Assuming L = 1 m, Q = (π * (0.05)^4 * 1000) / (8 * 0.1 * 1) = 0.01 m³/s.

Shear stress = viscosity × (velocity/radius) = 0.1 × (1/0.05) = 2 Pa.

Using the formula for shear stress, τ = (4 * η * Q) / (π * r^3), we find τ = 0.4 Pa.

Using the equation ω_f = ω + αt, where α = τ/I, we get ω_f = ω + (τ/I)t.

From Newton's second law for rotation, τ = Iα, thus α = τ/I.

From Newton's second law for rotation, τ = Iα, thus α = τ/I.

First convert rpm to rad/s: 1000 rpm = (1000 * 2π)/60 rad/s. Then use α = (ωf - ωi)/t = (0 - (1000 * 2π)/60) / 10.