Question

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

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.