Solid Geometry #1 - Questions and Answers

Question 1 A light wave has a 670 nm wavelength in air. Its wavelength in a transparent solid is 420 nm. a. What is the light speed in this substance? b. What is the light's frequency in the solid? Answer 1 Step 1 Index of refrection: n= c/v n ⇒ index of refraction of an optical material. c⇒ speed of light in space, v → speed of light in the material Light always travels more slowly in a material than in vacuum, so the value of n in anything other than vacuum is always greater than unity. Note that: the index of refraction depends on each material Apply: in most cases we need the speed of light in a specific medium, so we use the index of refraction for this medium to get the speed of light in this medium, Step 2 Wave speed for a periodic wave: v = λ/T = λf The wave advances a distance of one wavelength λ during one period T, so the wave speed is given by: v = λ/T = λ * (1/T) v = λf v ⇒ the speed of wave, f⇒ the frequency of the wave, T⇒ The periodic time, λ ⇒ wavelength of the wave. This equation is applied to any kind of periodic waves. The wave of a string will explain it more: For a periodic wave, the shape of the string at any instant is a repeating pattern. The wavelength of the wave is the distance from one crest to another, or from one trough to the next, or from any point to the corresponding point on the next repetition of the wave shape. The wave pattern travels with constant speed and advances a distance of one wavelength in a time interval of one period T. Apply: in most problems we use the equation directly to get one of the variables by knowing the other variables such as the wave speed and wavelength. Sometimes we need to get the frequency by knowing the time period by using the relation above or getting the speed using the frequency and the wavelength. Step 3 a) v = c = 3 × 10⁸ m/s v = c = λf f = c/λ = (3 × 10⁸ m/s) / (670 × 10⁻⁹ m) = (3 * 10⁸)/(670 * 10⁻⁹) = 4.478 * 10¹⁴ Hz ⇒ The frequency is the same in air and glass as is same source v_solid = f_solid * λ_solid v_solid = (4.478 × 10¹⁴ Hz) (420 × 10⁻⁹ m) = 1.88 × 10⁸ m/s Step 4 b) f_air = f_solid = f = c/λ = (3 × 10⁸ m/s)/(670 × 10⁻⁹ m) = 4.478 × 10¹⁴ Hz Answer 2 Step 1 Only the wavelength and speed of a beam of light change as it travels through different media, not its frequency. The experimental speed of light in air is: 3 × 10⁸ m/s Therefore, the frequency of this beam of light is: v = fλ; where v is speed, f is frequency, and λ is the wavelength. The frequency thus is: f = v/λ = (3 * 10⁸ m/s) / (670 × 10⁻⁹ m) = 4.48 × 10¹⁴ Hz The frequency will remain the same in the solid medium The speed will then become: 4.48 x 10¹⁴ * 420 x 10⁻⁹ = 1.88 × 10⁸ meters per second Question 2 How many faces does a rectangular prism have? Answer 1 A rectangular prism has six sides, two at each end and four around. Answer 2 6. For example, cube and cuboid are rectangular prisms each with 6 faces. Question 3 The volume of the trapezoidal prism is 1279.152 cubic feet. Determine the height of the trapezoid base. Answer The volume of a prism is the product of the base B and its height H: V=BH We are given that V=1279.152 cubic feet and H=25.2 ft so we can solve for B: 1279.152 = B(25.2) B = (1279.152/25.2) B = 50.76 ft² B is the area of the trapezoid base. The area of a trapezoid is given by: A = 1/2(b₁+b₂) height where b1 and b2 are the base lengths and h is the height. So, we have: 50.76 = 1/2(4.1 + 10)h 50.76 = 1/2(14.1)h 50.76 = 7.05h h = 50.76/7.05 h = 7.2 ft Question 4 Three vertices of parallelogram DEFG are D(-2, -4), F(0, 7), and G(1, 0). Find the coordinates of E. The coordinates of E are ______ Answer Plot the three points. We note that point E must be between D and F. By definition of parallelogram, opposite sides must be parallel so DE || FG and EF || DG. Since D is 3 units to the left and 4 units down of G, then E is also 3 units to the left and 4 units down of F so: [Image of a graph with D at (-2, -4), F at (0, 7) and G at (1, 0). E is plotted at (-3, 3)] E is at (-3, 3) Question 5 Equations should be written in cylindrical coordinates 3x + 2y + z = 6 Answer The given equation is 3x + 2y + z = 6 we know x = r cos θ, y = r sin θ and z=z ... we have 3r cos θ + 2r sin θ + z = 6 Result: 3r cos θ + 2r sin θ + z = 6