Let's denote the dimensions of the rectangular base of the box as follows: let the shorter side be
x and the longer side be
3x. Let the height of the box be
h. The volume of the box
V is fixed and given by the product of its dimensions:
V=xâ‹…3xâ‹…h=3x2hSince the volume is fixed, we can express
h in terms of
x and
V :
h=‌ Next, let's find the surface area
S of the box, which includes the area of the base, the top, and the four sides:
S=2â‹…(xâ‹…3x)+2â‹…(xâ‹…h)+2â‹…(3xâ‹…h)Simplifying, we get:
S=2â‹…3x2+2â‹…xh+2â‹…3xh=6x2+8xh We need to substitute
h from our previous expression:
S=6x2+8x(‌)Simplifying further:
S=6x2+‌ To find the most economical proportion, we need to minimize this surface area
S. To do that, we differentiate
S with respect to
x and set the derivative to zero:
‌‌=‌(6x2+‌)‌‌=12x−‌ Setting the derivative equal to zero for minimization:
‌12x−‌=0‌12x=‌‌36x3=8V‌x3=‌‌x3=‌‌x=(‌)1∕3 Now, substituting
x back to find
h :
‌h=‌=‌‌h=‌‌h=‌‌h=‌‌h=‌‌h=(‌)2∕3x Since the shorter side
x was taken as the base dimension, we conclude that the height
h is proportional to the shorter side of the base. Hence, the most economical proportion of the height of the covered box is:
‌ times the shorter side of the base.
Therefore, the correct answer is:
Option A
‌×‌ shorter side of base ‌