3. (a) Suppose AABC is a triangle on the unit sphere S and suppose A = π/4, B = π/3, and C= π/2. What are cos a, cos b and cos c? (b) Suppose AABC is a triangle on the pseudosphere H and suppose A = π/6, B = π/4, and C= π/2. What are cosh a, cosh b and cosh c?
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- 1.A square with one side along the y-axis is rotated 360° about the y-axis. What is the resulting solid? 2. A family is buying a house and wants to pay less than $150$150 per square foot. The price of one 1,8001,800 square foot house is $242,000.$242,000. Which statement is true? A. If the family purchases the house, the family will stay within the limit of $150 per square foot and will be under by about $15.56 per square foot. B. If the family purchases the house, the family will stay within the limit of $150 per square foot and will be under by about $25.56 per square foot. C.If the family purchases the house, the family will not stay within the limit of $150 per square foot and will be over by about $15.56 per square foot. D.If the family purchases the house, the family will not stay within the limit of $150 per square foot and will be over by about $16.67 per square foot.2. Give illustrations for the three - line and four - line geometries.2.11 Let A = p cos 9 ap + pz2 sin az (a) Transform A into rectangular coordinates and calculate its magnitude at point (3, -4 , 0). (b) Transform A into spherical system and calculate its magnitude at point (3, —4, 0).
- 5) Consider four points A, B, C, and D whose geometrical locations correspond to the corners of a square with sides of length 1 mm. Calculate the potential differences (in mV) VAB, VBA, VAC, VCA, VAD, VDA, VBC, VCB, VBD, VDB, VCD, VDC between the points in a uniform electric field of 3 V/m parallel to the two sides (and perpendicular to the other two) of the square. (Show your work)Find the areas of the regions . 1. Shared by the circle r = 2 and the cardioid r = 2(1 - cos u) 2. Shared by the cardioids r = 2(1 + cos u) and r = 2(1 - cos u)If you have two points and conic angles theta 1 and theta 2 have a difference of 180 degrees, how do you show the harmonic mean of repective values of r to be a const.
- Show that the path given by r(t) = (cos t,cos(2t), sint) intersects the xy-plane infinitely many times, but the underlying space curve intersects the xy-plane only twice.P.I.D as in Priniciple Ideal Domain.In math land, there is a clock. The hour hand of this clock is 1 ft long. However, the clock is weird, in that it takes precisely 2π hours for the hour hand to do a full rotation. Let’s position the clock in the xy-plane so that the center of the clock is at (0, 0), and the tip of the hour hand at "midnight" is at (0, 1). (a) It follows from the definition of sin and cos that at time t, the tip of the hour hand is at position (x, y) = (cos(t),sin(t)). Check that this makes sense by “manually” finding the position of the clock at time t = 0, π/2 , π, 3π/2 , and 2π. (b) The equation describing the circle that the hour hand traces out is x2 +y2 = 1. We’re going to calculate how fast the x-coordinate of the tip of the hour hand changes with time. We’ll do this in two ways. Using the fact that x = cos(t), what is dx/dt ? (c) Next, we’ll compute dx/dt in another way. Using the fact that y = sin(t), what is dy/dt ?
- In math land, there is a clock. The hour hand of this clock is 1 ft long. However, the clock is weird, in that it takes precisely 2π hours for the hour hand to do a full rotation. Let’s position the clock in the xy-plane so that the center of the clock is at (0, 0), and the tip of the hour hand at "midnight" is at (0, 1). (a) It follows from the definition of sin and cos that at time t, the tip of the hour hand is at position (x, y) = (cos(t),sin(t)). Check that this makes sense by “manually” finding the position of the clock at time t = 0, π/2 , π, 3π/2 , and 2π. (b) The equation describing the circle that the hour hand traces out is x2 +y2 = 1. We’re going to calculate how fast the x-coordinate of the tip of the hour hand changes with time. We’ll do this in two ways. Using the fact that x = cos(t), what is dx/dt ? (c) Next, we’ll compute dx/dt in another way. Using the fact that y = sin(t), what is dy/dt ? (d) Next, use implicit differentiation to find dx/dy . Express your answer…Suppose that U is a solution to the Laplace equation in the disk Ω = {r ≤ 1} andthat U(1, θ) = 5 − sin2θ.(i) Without finding the solution to the equation, compute the value of U at theorigin – i.e. at r = 0.(ii) Without finding the solution to the equation, determine the location of themaxima and minima of U in Ω.(Hint: sin2θ =(1−cos 2θ)/2.)1. Find the length of the curve r = cos³theta, 0≤ 0 ≤ π/4.