(e) (3 points) Without looking at the testing area, you allow an assistant to move the object to a new position. The sonar now shows that at 9 = 45 and 935 = 30, there is an object p = 8 motors away. Find the object's ($1 3/, z) coordinates. (d) (4 points) The maximum range of the detector is not uniform. At a given angle Q5 from the forward axis, the detector can only detect objects up to a distance of p = 200 cos Q6. Convert this equation to rectilinear (ac, y, z) coordinates. (f) (2 points) Your supervisor asks you to compute the total volume in cubic meters that the detector is able to cover. Unfortunately, you don't have the required knowledge to carry out the sphericalcoordinate doubleintegral this would require. Fortunately, you do have the ability to draw pictures, and it turns out that this volume is just a sum of two basic geometric shapes. Compute the volume the detector is able to cover. (e) (2 points) The detector also can't work with any possible angle; the max 7T irnum value of (b is 45, or Z radians. Convert this equation to recti- linear (sc, y= z) coordinates. HINT: First, take the cosine of both sides. Sonar is a method for detecting objects by bouncing sound off of them. A ping of sound (often of a very high frequency, ultrasound) is broadcast in a direction, and if it bounces, the time of the echo gives the distance to the object. (One must multiply by half the speed of sound, of course, but we won't worry about that for this problem.) You're testing out a new sonar system; your goal is to locate some test objects and mathematically describe the shape of the sonar's range. For the purpose of this question, the :raXis is to the right, the y-axis points down, and the zaxis points fomard. (This does not affect conversion equations, just the physical interpretation of them.) (a) (3 points) The initial test of the sonar system is to place an object 12 meters directly in front of the detector. In other words, the position of this object will be (2r,y,z) = (0,0,12). What are the corresponding spherical coordinates? The coordinates 6 and d) tell us where on the screen the object will show up, and ,0 tells us the detected distance. (b) (1 point) The test object is now moved 3 meters to the right and 4 meters up, to (39,9521) = (3, 4,12). If the sonar is working properly, what distance should it now show for the object? In other words, what is the new value of p