Homework SourseA rectangular storage container with an open top is to have
A rectangular storage container with an open top is to have a volume of 10m2 The length of its base is twice the width. Material for the base costs $10 per square meter. Material for the sides costs $6 per square meter. Find the cost of materials for the cheapest such container.
Solution
V= base *height = 10 Heavy action on the width! base =length *width length = l = 2w base= b in m²= l * w = 2w *w = 2w² perimeter (distance around base) =p= 2(w +l) = 2( w+2w); 2(3w)= 6w height = 10/b =10/2w² = 5/w² sides= h*p=side in m²= 5/w² * 6w = 30w/ w²= 30/w Cost, C = $10(base) +$6(sides) C = 10(2w²) + 6(30/w) C= 20w² +180/w C= 20w² +180 w?¹ To Minimize Cost, get C\' by setting to zero and diffrentiate 0 = 40w + - 180 w?² 180/w² =40w 180/40 = w³ 4.5 = w³ w = cube root 4.5 = 1.651 l= 2w = 2(1.651) =3.302 b= 2w or lw= 5.4514 sides= 30/w = 18.171 so cheapest cost: $10(5.4514) + $6(18.171)= 54.52 + 109.03= $163.55
