Unit 3 Performance Task Goal: Students will rewrite the America's Cup mle using radicals and evaluate the rule with the given values. Then students will solve an inequality as they design a yacht of different dimensions. Role: Yacht Designer Audience: Amenca's Cup fans Situation: America's Cup is the name of both an international yacht race and the trophy that is awarded to the winner. To compete in America's Cup, a yacht must meet certain standards of design and construction. The inequality below describes one of the nJIes for the yachts. 1 1 L + 1. 2551-7 9. 805 0.686 In the equality, L is the boat's length (in meters), 3 is the sail area {in square meters], and D is the displacement (in cubic meters}. Displacement is the volume of the boat that is underwater. Yacht designers try to make the value of the left side of the inequality as close to 24 as possible. 24 The table below lists data on the A045 Maj] Catamaran, a yacht that was built to compete in the 2-313 America's Cu p. You are going to design a new yacht that will satisfy the Americas Cup rule and have twice the length and twice the sail area 01 the A045. A845 _I_._ ._.I._ a Catamaran _13.45 meters 937 square meters _1.3 cubic meters Product: You are going to design a new yacht that will satisfy the America's Cup mle and have twice the length and twice the sail area of the A045. Then, verify that the A045 W Catamaran satises the America's Cup rule and nd all possible displacements for your new yacht. Attenvards, draw the design of the yacht. Indicate the dimensions from the stem to the bow. Select the sail design that will best represent your team's competitive spirit. 1} Looking back at the rule for America's Cup yachts: a) Write the expression on the left side of the rule in radicai form. b) What is the unit of measure associated with each variable it the expression? What is the unit associated with the expression as a whole? Explain. c) Estimate the value of the expression for the A045 W Catamaran. Show your work and explain how you chose values for the length, sait area, and displacement to make the calculation easier. 2} You wrote the expression on the left side of the Penelica's Cup rule in radical form [problem 1a}, now use this form of the expression to write a function n} that gives the value of the expression for the new yacht in terms of the displacement D. a) To write the function, what variables will you replace with specic values? What are those vatues? [1) Write and simplify the function. c) Describe how you could use the function to write and solve an equation to nd the displacement that makes the value of the left side of the America's Cup mle exactly 24. 3) You wrote a function o} for the new yacht {problem 2b]. The function gives the value of the expression on the left side of the America's Cup rule for any displacement 1). Use a graphing calculator {Desmos} to graph this function. Select all of the following that are two. Explain your reasoning. a} For any viewing window, the graph of n) lies entirely above the xaxis. b} In the context of this realwortd situation, the relevant domain of the function is all real numbers. c) As the value of D increases, the value of n) decreases. d} The graph of the function intersects the horizontal line y = 24 at exactly one point. e} The graph shows that when D = 10, n) is less than 24. f] The graph shows that a displacement of 3G cubic meters is one possible displacement for the new yacht. 4} Now solve the problem in the Task Description by verifying that the A034 W Catamaran satises the America's Cup rule, and by nding the possible displacement for your yacht. Show all your work and explain each step of your solution. 5) Draw the design of the yacht. Indicate the dimensions from the stem to the bow. Select the sail design that will best represent your team's competitive spirit. Standards: FlF.C.Tb Graph square root, cube root, and piecewisedened functions, including step functions and absolute value functions. FIF.C.8 Write a function dened by an expression in different but equivalent forms to reveal and explain different properties of the function. NRN.A.2 Rewrite expressions involving radicals and rational exponents using the properties of exponents. AREIAE Solve simple rational and radical equations in one variable, and give examples showing how extraneous solutions may arise. ACED.A.4 Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm's few 'v'= if? to nightlight resistance R