Question

the volume of cubic is increasing at a rate of 12 cubic centimeter per second how fast is the surface area increasing when the length of an edge is 15 cm ? chapter name application of derivation​

Answer 1

Answer:

Step-by-step explanation:

Let x be the length of a side, V be the volume, and S be the surface area of the cube.

Then, V=x  

3

  and S=6x  

2

 

It is given that  

dt

dV

​  

=8cm  

3

/s.

Then, by using the chain rule, we have:  

∴8=  

dt

dV

​  

=  

dt

d

​  

(x  

3

)⋅  

dx

d

​  

=3x  

2

⋅  

dt

dx

​  

 

⇒  

dt

dx

​  

=  

3x  

2

 

8

​  

.........(1)

Now,  

dt

dS

​  

=  

dt

d

​  

(6x  

2

)⋅  

dx

d

​  

=(12x)⋅  

dt

dx

​  

 [By chain rule]

=12x⋅  

dt

dx

​  

=12x⋅(  

3x  

2

 

8

​  

)=  

x

32

​  

 

Thus, when x=12 cm,    

dt

dS

​  

=  

12

32

​  

cm  

2

/s=  

3

8

​  

cm  

2

/s.

Hence, if the length of the edge of the cube is 12 cm, then the surface area is increasing  at the rate of  

3

8

​  

cm  

2

/s.