Homework SourseA parcel delivery service will accept boxes for delivery pro
A parcel delivery service will accept boxes for delivery provided the sum of the length, width, and height not exceed 120 inches. Suppose you wish to ship a box whose height is twice its width and has the maximum possible volume.
a)Let x represent the width, in inches of the box. Write a function V(x) that gives the volume of the box in terms of x
b)Determine all critical points for V(x) and find the dimensions of the box of maximum volume.
show work and clearly state conclusion
a)Let x represent the width, in inches of the box. Write a function V(x) that gives the volume of the box in terms of x
b)Determine all critical points for V(x) and find the dimensions of the box of maximum volume.
show work and clearly state conclusion
Solution
a) x*2x*(120-3x) = volume x = width 2x = height l is whatever is left of the 120 inches once width and height have been taken away. b) we want to maximize volume so let\'s multiply out the volume equation and then find where slope is zero, since this will be a critical point 240x^2 - 6x^3 = volume divide by six since it will be set as a ratio 40x^2 - x^3 80x -3x^2 = 0 x(80 - 3x) = 0 width x = 80/3 height = 160/3 length = 120/3 Height is longer than ideal since it has to be 2x width is shorter than ideal since it has to be half height length is ideal Ideal would be a cube with each side 120/3. This is as close to a cube as possible given the constraints critical points, x = 0 is a critical point, volume is zero. another critical point would be width is 40 and height is 80 and volume is also zero.
