Question

A rectangular tank with a square base and no top is to have a volume of 10 m3 . Material for the bottom of the tank costs $15/m2 and material for the sides costs $6/m2

a. Find the dimensions of the cheapest such tank that can be constructed.

b. How much would the tank in part a. cost to build?

Answer 1

!st we have to find cost of tank in terms of single variable and then minimize it by using derivative

For relative maximum or minimum

To maximize or minimize a function f(x) 1st we need to find its critical points

For critical points we need to put differentiation of f(x) equal to zero i.e. f '(x) = 0

Then check second derivative of f(x) i.e. f ''(x) is positive or negative at critical numbers If f ''(x) >0 then that gives minimum value of f(x) and if f ''(x) <0 then that give maximum value of f(x)

her base is sancore on base is sancse on sides non and height you Volsone or hok a nroxy=5a. Do ny=10 (siver) => y lo -0. Area ob boltom an as oratorial Cast bor bottom= $15% Cost of botton ci a noise Ara o aside - hoy, As matosial cost corsides &6/ou? Cast al sides erano 20o xo=2200 & top cost e-citer Ca 150%+24 ny replaciz y bg coon ③ c=150%+ 260. Lo - 1500 + 240 © we have to minimize C. difleren hale e with resped ton, den 15. dan (n) +250. In (1)

[an (an) - 5000] dena 15xz2 +240 (-2-2) 2 pon - 200- _ At cortical vakes. Bon- 2on2=0 yon = 250 s n=240 = 2 8=23 n=2 Now are = 30 d (0) - 240dn (1-2) - 30-2600 (-2003) [Bon du latinsy are (as n=2) = 30 + 480 - 30+60=5070 As are so cosmisinm. ler cheapest kak n=2, yz 12 4 = 2.som For cheapest box height = 2.5m and lessth or side of bose a 2m. none 6) Forminium cost put n-2 in ② c=15+ 2 + 220 - 60 +120=1808 Cast or cheapest book - 1804