Probability Questions

1. The time to failure t. in hours, of a certain machine can often be assumed to be exponentially distributed with probability density function f(t) = (1/89)e (-t/89), OWhat is the probability that a failure will occur in 49 hours or less? 2. The rate of change in a person's body temperature, with respect to the dosage of x milligrams of a drug, is given by D'(x) 2/(x+6). One milligram raises the temperature 2.2C. Find the function giving the total temperature change. 3. The rate of growth of the profit (in millions) from an invention is approximated by P(x) xe^(-x^2) where x represents time measured in years. The total profit in year 2 that the invention is in operation is $10,000. Find the total profit function. Round to three decimal places where appropriate. 4. The total revenue from the sale of x stereos is given by R(x) = 2000(1 - Find the marginal average revenue. 5. A hotel has 270 units. All rooms are occupied when the hotel charges $90 per day for a room. For every increase of x dollars in the daily room rate, there are x rooms vacant. Each occupied room costs $28 per day to service and maintain. Find what the hotel charge per day should be in order to maximize daily profit. 6. S(x) = x3 9x2 + 165x + 1300, 5 5 x 20 is an approximation to the number of salmon swimming upstream to spawn, where x represents the water temperature in degrees Celsius. Find the temperature that produces the maximum number of salmon. 7. The population of students at a university is given by the formula where t represents the time in years since 2000. Find the rate of change of the student population in both 2006 and 2016. 8. The force F (in N) exerted by a cam on a lever is given by F(x) =x*-13x+44x-55x+20, 1 x 5, where x is the distance (in cm) from the center of rotation of the cam to the edge of the cam in contact with the lever. Find the instantaneous rate of change of F with respect to x when x = cm. 9. Numbers is a game where you bet $1.00 on any three-digit number from 000 to 999. If your number comes up, you get $600.00. Find the expected payback. 10. Sketch the graph of f(x)=x-3x'22-4 and show all extrema, inflection points, and asymptotes, where applicable. Please show and explain all of your conclusions.