At a certain instant the edge of a cube is 7 inches long and the volume is increasing at the rate of 3 cubic inches per minute. How fast is the surface area of the cube increasing

The surface area increases at 12/7 square inches per minute

Step-by-step explanation:

Given

--- side length

--- Rate at which volume increases

Required

The rate at the area

The volume of a cube is:

Differentiate with respect to time

When and

The expression becomes:

Divide both sides by 3

Divide both sides by 49

The surface area is calculated as:

Differentiate with respect to time

Substitute: and

The surface area increases at 12/7 square inches per minute

hope it you

step-by-step explanation:

sin(7pi/6 - pi/3)  

= sin(7pi/6 - 2pi/6)  

= sin(5pi/6)  

= sin(pi - pi/6)  

= sin(pi/6)  

= 1/2  

sin 7pi/6 - sin pi/3  

= sin (pi + pi/6) - sin pi/3  

= -sin pi/6 - sin pi/3  

= -1/2 - sqrt(3)/2  

= -(1+sqrt(3))2

we have three similar triangles, because each has a right angle and shares an angle.   let's write the angles in order: opposite to short leg, long leg, hypotenuse.

cab similar to bad similar to cbd

or as ratios,

ca: ab: cb = ba: ad: bd = cb: bd: cd

we also know

ac = ad + cd

(ad+cd): ab: cb = ba: ad: bd = cb: bd: cd

(ad+cd)/cb=cb/cd

we have cb=6, ad=5 and seek x=cd.

we reject the negative root and conclude x=4

answer: 4