Optimization
Practice
2.1 The Volume of a can is 9π in³. The top and bottom cost twice as much to manufacture as the side. The lateral surface costs 1/4 cent per in². The top and bottom cost 1/2 cent per in². Find the cylinder can of minimum cost.
2.2 An upside down cone is inscribed in a hemisphere. The radius of the hemisphere is 6 inches. Find the dimensions and volume of the cone of greatest volume.
2.3 Find the dimensions of the rectangle of the largest area with sides parallel to the coordinate axes that can be inscribed in the region enclosed b the graphs of:
2.4 Given a right triangle with a hypotenuse of 10cm, let θ = one of the acute angles.
a) For what value of θ (in radians) is the perimeter a maximum?
Let x = side adjacent to θ and let y = side across from θ.
Perimeter = 10 + y + x
Throw out the 
because it is larger than 90 degrees so can’t be in the triangle.
b) What is the maximum perimeter?
c) Find the minimum perimeter.
Conceptual: when the triangle is as close to flat as possible. The perimeter will be just over 20cm.
