A rectangular box with no top (pictured above) is to contain 2250 cubic inches of volume. Find the dimensions of the box that will minimize the surface area. The length (l) of the base is three times the width (w).

Answer:Height = 7.5 inchesWidth = 10 inchesLength = 30 inchesStep-by-step explanation:Volume of the shape (V) = length (l) * width (w) * height (h)From the question, the length = 3wSo, V = 3w * w * hV = 3w²hh = V/3w²h = 2250/3w²Therefore h=V/(3w^2) where V is the volume.Considering the fact that the top of the rectangular box is missingThe Surface Area (S) = (l*w) + 2(l*h) + 2(w*h) --------------------- (replace l with 3w and h with 2250/3w²)S = 3w*w + 2(3w)(2250/3w²) + 2w(2250/3w²)S = 3w² + 4500/w² + 1500/wS = 3w² + 6000/wWe need to find the minimum of the above function (equation) by finding its first derivative with respect to ADifferentiating the above function, we have0 = 6w - 6000/w²6000/w² = 6w6000 = 6w³ ------------ Divide through by 61000 = w³ ---------------- Find the cube root of both sides10 = ww = 10Hence, width of the rectangular box is 10 inchesTo solve height, we useh = 2250/3w²So, h = 2250/(3*10²)h = 2250/300h = 7.5 inchesTo find the length, we use l = 3wSo, l = 3 * 10l = 30 inches