In the figure, TU is tangent to the circle at point U. Use the figure to answer the questions.

Describe the relationship among the lengths of the segments formed by the secant, RT , and the tangent segment, TU . You may use words and/or an equation. (2 points)
Suppose RT=25 in. and ST=4 in. Is it possible to find the length of ? If so, show how to find the length. If not, explain why not. (3 points)

Find the arc length parameter along the given curve from the point where tequals=0 by evaluating the integral s(t)equals=integral from 0 to t startabsolutevalue bold v left parenthesis tau right parenthesis endabsolutevalue d tau∫0tv(τ) dτ. then find the length of the indicated portion of the curve r(t)equals=1010cosine tcost iplus+1010sine tsint jplus+88t k, where 0less than or equals≤tless than or equals≤startfraction pi over 3 endfraction π 3.

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."

Celeste wants to have her hair cut and permed and also go to lunch. she knows she will need $50. the perm cost twice as much as her haircut and she needs $5 for lunch. how much does the perm cost?