Let the lengths be 5x and 7x. Given 5x + 7x = 72, 12x = 72, x = 6. Therefore, shorter rope = 5x = 5*6 = 30.

If the shorter side is 9 cm and the ratio is 3:4, then the longer side is (4/3) × 9 cm = 12 cm.

Let the sides be 3x and 5x. Then, 3x + 5x = 64, 8x = 64, x = 8. Therefore, the longer side = 5x = 5*8 = 40.

Let the lengths of the sides be 7x and 5x. Then, 7x + 5x = 48, which gives 12x = 48. Thus, x = 4, and the longer side is 7x = 28 cm.

Let the shorter side be 7x and the longer side be 9x. Given 9x = 36, x = 4. Therefore, shorter side = 7x = 7*4 = 28.

The ratio of cats to dogs is 2:3. If there are 30 dogs, the number of cats can be calculated as (2/3) * 30 = 20 cats.

Let the sides be 3x, 4x, and 5x. Then, 3x + 4x + 5x = 36. Thus, 12x = 36, giving x = 3. The longest side is 5x = 15 cm.

If the shortest side is 6 cm, the sides are 6, 8, and 10 cm. Perimeter = 6 + 8 + 10 = 24 cm.

A triangle with sides in the ratio 3:4:5 is a right-angled triangle.

Let speeds be 5x and 7x. Given 7x = 140, x = 20. Slower car's speed = 5x = 5*20 = 100 km.

Let the numbers be 7x and 9x. Then, 7x + 9x = 128, 16x = 128, x = 8. Therefore, the numbers are 7*8 = 56 and 9*8 = 72.

Let the numbers be 7x and 9x. Then, 7x + 9x = 128, 16x = 128, x = 8. Larger number = 9x = 9*8 = 72.

Critical angle θc = sin⁻¹(1/1.33) ≈ 48.6°.

The critical angle θc can be calculated as θc = sin^(-1)(n2/n1) = sin^(-1)(1.00/1.33) ≈ 48.6°.

Speed of light in medium = c/n = (3 x 10^8 m/s) / 1.33 ≈ 2.25 x 10^8 m/s.

The critical angle θc is given by sin(θc) = n2/n1 = 1/1.5, which gives θc ≈ 41.8°.

The speed of light in a medium is given by v = c/n. Here, c = 3 x 10^8 m/s and n = 1.5, so v = 3 x 10^8 / 1.5 = 2 x 10^8 m/s.

Speed of light in medium = c/n = (3 x 10^8 m/s) / 1.5 = 2 x 10^8 m/s.

Wavelength in medium = λ/v = λ0/n = 600 nm / 1.5 = 400 nm.

Using the formula θc = sin^(-1)(1/n), where n = 2.0, we find θc = sin^(-1)(1/2) = 30°.

Using the formula sin(θc) = n2/n1, where n1 = 2.0 (medium) and n2 = 1.0 (air), we find sin(θc) = 1.0/2.0 = 0.5, leading to θc = 60°.

Using the formula sin(θc) = n2/n1, we have sin(θc) = 1.00/2.00, leading to θc ≈ 30°.

Critical angle θc = sin⁻¹(n2/n1) = sin⁻¹(1.00/2.00) ≈ 30°; thus, maximum angle of incidence for total internal reflection is 60°.

The speed of light in a medium is given by v = c/n, where n is the refractive index. If n > 1, the speed decreases.

The speed of light in a medium is given by v = c/n, where n is the refractive index. If n > 1, then v < c, meaning the speed decreases.

The speed of light in a medium is less than that in vacuum, given by v = c/n, where n > 1.

The wavelength of light decreases in a medium with a refractive index greater than 1, as λ' = λ/n.

The wavelength of light decreases in a medium with a refractive index greater than 1, as it is given by λ' = λ/n.

When light reflects off a medium with a higher refractive index, it undergoes a phase change of π (180 degrees).

Using the formula sin(θc) = n2/n1, where n1 = 2.42 (diamond) and n2 = 1.00 (air), we find θc ≈ 24.4°.