contestada

a box of volume 252 m3 with a square bottom and no top is made of two different materials. the cost of the bottom is $40/m2 and the cost of the sides is $30/m2. find the dimensions of the box that minimize the total cost.

Respuesta :

The dimensions of the box that minimizes the total cost is side of the square base is 7.23 m and height of the box is 4.82 m   .

In the question ,

it is given that ,

the bottom of the box is =  square ,

let the side of the square bottom be = "s" ,

so , the base area [tex]=[/tex] s²

let the height of the box [tex]=[/tex] h ,

Volume of the box = height × (base area)

So , the Volume of the box [tex]=[/tex] s²h

given that the volume of the box is 252 m³ , that means

s²h = 252

h = 252/s²

given that the Cost of bottom = $40 per m²

and the Cost of sides = $30 per m²

So , the total cost = 40s² + 30×(4sh)

substituting the value of h= 252/s² , we get

C = 40s² + 120s×252/s²

C = 40s² + 30240/s

to minimize the cost we differentiate with respect to s , and equating it  to 0 ,

we get ,

80s - 30240/s² = 0

s³ = 30240/80

s³ = 378

s = 7.23
again differentiating C = 40s² + 30240/s with respect to s , and equating it  to 0 ,

we get ,

d²C/ds² = 80 + (30240*2)/s³

at s=7.23 , d²C/ds² > 0

So , the minimum is at s = 7.23 ,

we have h = 252/(7.23)²

h = 4.82 .

Therefore , The dimensions of the box that minimizes the total cost is side of the square base is 7.23 m and height of the box is 4.82 m   .  

Learn more about Volume here

https://brainly.com/question/15707980

#SPJ4

Otras preguntas