Arain gutter is to be constructed of aluminum sheets 12 inches wide. after marking off a length of 4 inches from each edge, this length is bent up at an angle θ. the area a of the opening may be expressed as the function: a(θ) = 16 sin θ ⋅ (cos θ + 1). if θ = 45°, what is the area of the opening?

What is the value of x? enter your answer in the box. photo attached.

Asays "we are both knaves" and b says nothing. exercises 24–31 relate to inhabitants of an island on which there are three kinds of people: knights who always tell the truth, knaves who always lie, and spies (called normals by smullyan [sm78]) who can either lie or tell the truth. you encounter three people, a, b, and c. you know one of these people is a knight, one is a knave, and one is a spy. each of the three people knows the type of person each of other two is. for each of these situations, if possible, determine whether there is a unique solution and determine who the knave, knight, and spy are. when there is no unique solution, list all possible solutions or state that there are no solutions. 24. a says "c is the knave," b says, "a is the knight," and c says "i am the spy."

What is the missing step in this proof ?