Smith is given a choice between two sealed envelopes. one of them contains twice asmuch money as the other. since the money is in the form of checks payable to her, she cannot tell the envelopes apart before opening them. she picks one and finds a check for xamount of money. she is then given the choice to switch envelopes. now she reasons as so: “if i was lucky, then the other envelope contains ½ x; but if i was unlucky, then it contains 2x. each these scenarios has probability ½. so the expected utility, if i switch, is ½ × ½ x+ ½ × 2x= 5x/4. so i should switch.” yet this would meanthat whichever envelope one chooses in this situation, one should expect the other to contain more money. where does smith’s reasoning go wrong?

After a dreary day of rain, the sun peeks through the clouds and a rainbow forms. you notice the rainbow is the shape of a parabola. the equation for this parabola is y = -x2 + 36. graph of a parabola opening down at the vertex 0 comma 36 crossing the x–axis at negative 6 comma 0 and 6 comma 0. in the distance, an airplane is taking off. as it ascends during take-off, it makes a slanted line that cuts through the rainbow at two points. create a table of at least four values for the function that includes two points of intersection between the airplane and the rainbow. analyze the two functions. answer the following reflection questions in complete sentences. what is the domain and range of the rainbow? explain what the domain and range represent. do all of the values make sense in this situation? why or why not? what are the x- and y-intercepts of the rainbow? explain what each intercept represents. is the linear function you created with your table positive or negative? explain. what are the solutions or solution to the system of equations created? explain what it or they represent. create your own piecewise function with at least two functions. explain, using complete sentences, the steps for graphing the function. graph the function by hand or using a graphing software of your choice (remember to submit the graph).