Question

All boxes with a square base, an open top, and a volume of 200 ftº have a surface area given by S(x)=x2 + where x is the length of the sides of the base. Find the absolute minimum of the surface area function on the interval (0,00). What are the dimensions of the box with minimum surface area? Determine the derivative of the given function S(x). S'(x)=0 The absolute minimum value of the surface area function ist? (Round to three decimal places as needed.) The dimensions of the box with minimum surface area are a base of length it and a height of ft. (Round to three decimal places as needed.)

Answer 1

Let bong w length s height y volumea So 200 ny - 200 [ because box is square in shope S(n)= 1² + 800 o find Sin diff Sul on both side with S (W)=d 2 ] da [na + Roo -1/2 urso 22 To determine critical point S (nl=0 2n- 800 So nt 2n- Soo W - 400 In 7.368

To determins of 7.6368 surface Oreo is minironm destal dhe A 7368 dr Soy- & (2 x 800) dne dx = 2 + 1600 - 2 + 1600 7.3688 = 2x1600 yoo ot no 7.638 surface aree is miniwm S(7.3680= 77.3688 + 800 7768 = 54.288 & 108.5776 = 162. 8 656 absolute = 162.866 minimum volue of our face area is (162.866 ft

of a 7.368 ne ye = 368 200 y = 2368200 (7.360)" Y = 3.684 Dimension of box with minimum surface are a base of length & is 17.368 / ft and height is 13.684 PA