EXAMPLE 1 A rectangular box without a lid is to be made from 48 m2 of cardboard. Find the maximum volume of such a box. SOLUTION We let x, y, and z to be the length, width, and height, respectively, of the box in meters. Then we wish to maximize V = xyz subject to the constraint g(x, y, z) = 2xz + 2yz + xy = 48. Using the method of Lagrange multipliers, we look for the values of x, y, z, and λ such that ∇V = λ∇g and g(x, y, z) = 48. This gives the equations vx = λgx vy = λgy vz = λgz 2xz + 2yz + xy = 48

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.

25 ! the line of best fit is h = 5a + 86. predict the height at age 16.

The u-drive rent-a-truck company plans to spend $13 million on 320 new vehicles. each commercial van will cost $25 comma 000, each small truck $50 comma 000, and each large truck  $80 comma 000. past experience shows that they need twice as many vans as small trucks. how many of each type of vehicle can they buy?