Please answer exactly what the professor is asking. and show the work. :)

Distance and Midpoint Formulas and the Pythagorean Theorem Distance Formula Midpoint Formula Pythagorean Theorem + + d = ./(x2 x02 + (yz m2 (x1 2 x71 2 y.) a2 + b2 = c2 Star Trek Holodeck: 1. In a holodeck simulation, the Starship Enterprise at point A is traveling to meet the Excelsior Transport Ship at point B. The ship at point B is transporting an alien ambassador who must board the Enterprise but cannot use the transporter. a) The ships have agreed to meet at the middle of the line segment connecting the two ships' current positions. Which of the formulas above is useful? Use this formula to find the grid coordinates of the meeting location. Show your work. b) Find the holodeck distance between the ships' current positions, rounding to the nearest hundredth of a grid unit. Show your work. 1 grid unit = 300 astronomical units (AU) 1 grid unit = 300 astronomical units (AU) c) The distance found in part (b) is measured in grid units. If each grid unit in the holodeck represents 300 astronomical units (AU), find the distance to the nearest astronomical unit between the ships' current positions. Use your rounded answer from (b). The Pythagorean Theorem states that the square of the length of the hypotenuse of a right triangle is equal to the sum of the squares of the lengths ofthe two remaining sides. It is important to remember that the theorem applies only to right triangles. It is also helpful to note that c (the hypotenuse! must be the longest side. Measuring the depth of a lake: 2. You need to measure the depth of a large lake. Since the sonar equipment is very expensive, you decide to use your friend's boat. The boat has an anchor on a 50 ft line. You take the boat out to the middle of the lake on a windy day. You drop the anchor and let the wind push the boat until the anchor line is tight. Your GPS tells you that the boat has moved 40 feet. Assuming the bottom of the lake is flat, what is the depth of the lake? Show your work. Measuring the depth of a lake: 2. You need to measure the depth of a large lake. Since the sonar equipment is very expensive, you decide to use your friend's boat. The boat has an anchor on a 50 ft line. You take the boat out to the middle of the lake on a windy day. You drop the anchor and let the wind push the boat until the anchor line is tight. Your GPS tells you that the boat has moved 40 feet. Assuming the bottom of the lake is flat, what is the depth of the lake? Show your work. Cutting down a tree: 3. You need to remove a tree from your front yard. You have climbed the tree and trimmed off the branches, and you are now ready to chop down the trunk. It is very important that the tree falls in the right direction... away from your house. You have attached a wire to the top of the trunk. You will attach the other end ofthe wire to the ground and pull it tightly so that when the trunk begins to fall, the wire will pull it in the right direction. The trunk is 30 feet tall and you are going to attach the wire to the ground 12 feet away from the base of the trunk. a) How long must your wire be to reach the ground? Simplify your answer (this is an EXACT answer, so breakdown your radicand and bring any numbers outside the radical that you can), then round to the hundredth. Show your work. b) What assumption about the tree and the ground did we need to make here? Why did we need to make it? (Hint: What formula did you use, and what has to be true in order for you to be able to use it?) Is the foundation square? 4. You have hired a contractor to build a house. The rectangularfoundation (the concrete slab that the house is built on) has just been finished. You want to make sure that the construction is done well and the foundation is square, so you decide to see if the corners are right angles. You measure the width of the foundation to be 51 feet, the length to be 68 feet and the diagonal to be 85 feet. Did the company build the foundation with right angles? Prove your answer by showing work