Question

the cost of box, if rate of making the box is rs. 5 per cm², then volume is maximum is ​

Answer 1

Answer:

Since volume of the box =1024cm

3

let length of the side of square base be x cm and height of the box be y cm

∴ volume of box (V) =x

2

×y=1024

Since x

2

y=1024⇒y=

x

2

1024

let C denotes the cost of the box

∴C=2x

2

×5+4xy×2.50

=10x

2

+10xy=10x(x+y)

=10x(x+

x

2

1024

)

=

x

10

(x

3

+1024)

⇒C=10x

2

+

x

10240

...(i)

On differentiating both sides w.r.t. x we get

dx

dC

=20x+10240(−x)

−2

dx

dC

=20x−

x

2

10240

...(ii)

Now,

dx

dC

=0

⇒20x=

x

2

10240

⇒20x

3

=10240

⇒x

3

=512=8

3

⇒x=8

Again differentiating eq (ii) w.r.t. x we get

dx

2

d

2

C

=20−10240(−2).

x

3

1

dx

2

d

2

C

=20+

x

3

20480

At x=8

∴(

dx

2

d

2

C

)

x=8

=20+

512

2080

=60>0

For x=8 cost is minimum and the corresponding least cost of the box is:

C(8)=10.8

2

+

8

10240

=640+1280=1920

∴ least cost =Rs 1920