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05:03Folded boxes
a. Squares with sides of length x are cut out of each corner of a rectangular piece of cardboard measuring 5 ft by 8 ft. The resulting piece of cardboard is then folded into a box without a lid. Find the volume of the largest box that can be formed in this way.
Optimal soda can
a. Classical problem Find the radius and height of a cylindrical soda can with a volume of 354 cm³ that minimize the surface area.
Maximizing profit Suppose a tour guide has a bus that holds a maximum of 100 people. Assume his profit (in dollars) for taking people on a city tour is P(n) = n(50 - 0.5n) - 100. (Although P is defined only for positive integers, treat it as a continuous function.)
a. How many people should the guide take on a tour to maximize the profit?
Values of related functions Suppose f is differentiable on (-∞,∞) and assume it has a local extreme value at the point x = 2, where f(2) = 0. Let g(x) = xf(x) + 1 and let h(x) = xf(x) + x +1, for all values of x.
a. Evaluate g(2), h(2), g'(2), and h'(2).
Sketch a graph of a function f with the following properties.
f' < 0 and f" < 0, for x < 3
Suppose the objective function P= xy is subject to the constraint 10x + y = 100, where x and y are real numbers.
a. Eliminate the variable y from the objective function so that P is expressed as a function of one variable x.
