An insurance company offers the following insurance package to a customer with three businesses on the same street. The insurer will pay for all damages to the business that incurs the most damage, but nothing for the other businesses. Each business is worth a total of $4 million. The density functions are as follows, with values given in millions of dollars. Assume X, Y, and Z are independent random variables. f (x) is uniform on (0, 4). Find the probability density function of the amount the insurer will pay.

For every corresponding pair of cross sections, the area of the cross section of a sphere with radius r is equal to the area of the cross section of a cylinder with radius and height 2r minus the volume of two cones, each with a radius and height of r. a cross section of the sphere is and a cross section of the cylinder minus the cones, taken parallel to the base of cylinder, is the volume of the cylinder with radius r and height 2r is and the volume of each cone with radius r and height r is 1/3 pie r^3. so the volume of the cylinder minus the two cones is therefore, the volume of the cylinder is 4/3pie r^3 by cavalieri's principle. (fill in options are: r/2- r- 2r- an annulus- a circle -1/3pier^3- 2/3pier^3- 4/3pier^3- 5/3pier^3- 2pier^3- 4pier^3)