:your friends’ daughter madison has recently learned to spell some simple words. to encourage this, her parents got her a colorful set of refrigerator magnets featuring the letters of the alphabet (some number of copies of the letter a, some number of copies of letter b, and so on). the last time you saw her, the two of you spent some time arranging the magnets to spell out words that she knows. somehow with you and madison, things always end up getting more elaborate than originally planned, and soon the two of you were trying to spell out words so as to use up all magnets in the full set – that is, picking words that she knows how to spell, so that once they were all spelled out, each magnet was participating in the spelling of exactly one of the words. (multiple copies of words are okay here.) this turned out to be pretty difficult, and it was only later that you realized a plausible reason for this. suppose we consider a general version of the problem "using up all the refrigerator magnets", where we replace the english alphabet by an arbitrary collection of symbols, and we model madison’s vocabulary as an arbitrary set of strings over this collection of symbols. prove that determining if the collection of symbols can be used up to form strings from the vocabulary set is an np-complete problem.

2. fabulously fit offers memberships for$35 per month plus a $50 enrollmentfee. the fitness studio offersmemberships for $40 per month plus a$35 enrollment fee. in how many monthswill the fitness clubs cost the same? what will the cost be?

What are the 12 colors of the spectrum called?

Write a program that displays the following menu: geometry calculator 1. calculate the area of a circle 2. calculate the area of a rectangle 3. calculate the area of a triangle 4. quit enter your choice (1-4): if the user enters 1, the program should ask for the radius of the circle and then display its area. use the following formula: area = ď€(the square of r) use 3.14159 for ď€ and the radius of the circle for r. if the user enters 2, the program should ask for the length and width of the rectangle and then display the rectangle’s area. use the following formula: area = length * width if the user enters 3, the program should ask for the length of the triangle’s base and its height, and then display its area. use the following formula: area = base * height * .5 if the user enters 4, the program should end. input validation: display an error message if the user enters a number outside the range of 1 through 4 when selecting an item from the menu. do not accept negative values for the circle’s radius, the rectangle’s length or width, or the triangle’s base or height. note: if the user enters an improper menu choice (1-4), the program prints "the valid choices are 1 through 4. run the program again and select one of those." if the user enters a negative radius, the program prints "the radius can not be less than zero." if the user enters a negative value for height or base, the program prints "only enter positive values for base and height."