Consider the differential equationy'' − y' − 30y = 0.Verify that the functions e−5x and e6x form a fundamental set of solutions of the differential equation on the interval (−[infinity], [infinity]).The functions satisfy the differential equation and are linearly independent since the Wronskian We−5x, e6x = ≠ 0 for −[infinity] < x < [infinity].Form the general solution.

If x + 10 = 70, what is x? a. 40 b. 50 c. 60 d. 70 e. 80

Browning labs is testing a new growth inhibitor for a certain type of bacteria. the bacteria naturally grows exponentially each hour at a rate of 6.2%. the researchers know that the inhibitor will make the growth rate of the bacteria less than or equal to its natural growth rate. the sample currently contains 100 bacteria.the container holding the sample can hold only 300 bacteria, after which the sample will no longer grow. however, the researchers are increasing the size of the container at a constant rate allowing the container to hold 100 more bacteria each hour. they would like to determine the possible number of bacteria in the container over time.create a system of inequalities to model the situation above, and use it to determine how many of the solutions are viable.

Rockwell hardness of pins of a certain type is known to have a mean value of 50 and a standard deviation of 1.1. (round your answers to four decimal places.) (a) if the distribution is normal, what is the probability that the sample mean hardness for a random sample of 8 pins is at least 51?