A typical dish antenna is built as a surface of revolution obtained by revolving a parabola about an axis of symmetry. One of the main benefits of this design is that the resulting antenna exhibits very high gains in the direction towards which it points, making it well-suited for applications in which a strong directionality is needed-such as TV reception and radar. We can model such a parabolic antenna as the surface of revolution obtained by revolving the function 1/2 ㅠ O / [(1 + KR²) ¹/² − 1] K 퓸 [(1 2π 3K about the y-axis. Here R is the radius of the antenna, and K-the curvature at the tip-controls how flat it is. Compute the surface area of this antenna in terms of the parameters R and K. y = 1 +2RK)¹/² − 1] 2π 3K O 4√2 R³/2 K-1/4 3 [(1 + KR²) ³/2 − 1] [(₁1 (1+2RK)³/2 O 2√2 R¹/2 K-3/4 3 K - 1] _x²³, 0 ≤ x ≤ R

(a) Find three parametrizations that give the same graph as the given equation. (b) Find three parametrizations that give only a portion of the graph of the given equation.y = x2

A cyclist is travelling in a straight path at a constant speed along a dirt road with a button stuck to the outside of one of their wheels. The movement of the button (as viewed from the side) can be described using the equations of a cycloid, — r(t — sin t), у %3 г(1 — сo st) x = where t represents the angle through which the wheel has rotated and r represents the radius of the wheel. (a) Every time the button touches the ground, it leaves a mark on the dirt. If the marks are 64T cm apart, what is the radius of the tire wheel? (b) What value(s) of t correspond to the button touching the ground? dx (c) Show that when t = 2T, we have 0. Does this mean that the cyclist is stopped? Why or why not?

A golf ball has been hit off of the tee at an angle of elevation of 30 degrees and an initial velocity of 128 ft/sec. Using the helpful parametric models for projectile motion answer the following questions. x (t) = vo cos(0)t y (t) = hovo sin(0)t - 16t² a.) How long is the ball in the air (hang time)? b.) What is the maximum height of the ball? c.) How far, horizontally, does the ball travel in the air? (round to the nearest foot)