Find the maximum volume of a box that can be made by cutting out squares from the corners of an 8-inch by 15-inch rectangular sheet of cardboard and folding up the sides.

Answers: V = (15-2x)(8-2x)x V = 120x -46x²+ 4x³ V' = 120 -92x + 12x² 0 = 120 -92x + 12x² x = (92±52)/24 x = 6 or 5/3 Pick x = 5/3 Volume = (15-2*5/3)(8-2*5/3)*5/3 = 90.74 in³ To justify x = 5/3 is the max V" = -92 + 24x V"(5/3) = -92 + 24(5/3) <0 negative. Volume V(5/3) is max because the function is concave down with V"<0.

The answer is $9.90 with rounding to the nearest tenth