Question

a close box with a square base is to have a vlume of 1000cu.m. the material for the top and bottom of the box is to cost P100/sq.m. and the material for the sides is to cost twice the top. find the dimensions of the box so that the total cost of the material is minimum..?

Answer 1

say the side of the square base is \(\large x\)
so the area of bottom = area of top = \(\large ?\)

Answer 2

right..

Answer 3

x^2

Answer 4

yes and since `the material for the top and bottom of the box is to cost P100/sq.m. `

cost of top and bottom = \(\large (2x^2)\times 100\)

Answer 5

therefore it becomes 200x^2 right..?

Answer 6

yes thats the expression for cost of top and bottom,
lets work the lateral sides

Answer 7

say the height of box is \(h\)
since the box has four lateral sides, area of sides = \(4(x\times h) = 4xh \)

Answer 8

lets work again from this step

Answer 9

by the way why do we set the derivative equal to zero before?

Answer 10

` and the material for the sides is to cost twice the top.`

that means the cost of sides = \(\large 4xh \times 200 = 800xh\)

Answer 11

So the total cost would be : \(200x^2 + 800xh\)

Answer 12

see the mistake we committed earlier ?

Answer 13

ok

Answer 14

use the volume information and eliminate \(h\) like we did earlier :

\(x^2h = 1000 \implies h = \dfrac{1000}{x^2} \)

Answer 15

cost \(200x^2 + 800xh = 200x^2 + 800x \dfrac{1000}{x^2}\)

\(= 200x^2 + \dfrac{800000}{x} \)

Answer 16

differentiate this function
set it equal to 0 and solve x

Answer 17

back to my question earlier why did we equate it to zero...?

Answer 18

nevermind that question.. i think its the rule to get the value of x... any way thanks a lot.. you really are great....

Answer 19

nope, thats a very good question actually. lets see why setting the derivative equal to 0 works

Answer 20

you familiar with parabolas , right ?