- #1

- 35

- 0

## Homework Statement

The side of a cube increases at 1 cm / s. How fast is the

**diagonal**of the cube changing when the side is 1 cm?

## Homework Equations

Involves:

[itex] a^2+b^2=c^2 [/itex]

**Implicit Differentiation**

**Derivation**

## The Attempt at a Solution

I'm attempting to find the diagonal of the cube through Pythagorean Theory where I find the diagonal of

**one**side of the cube to give me an equation for side length 'A', after finding side length 'A' I know that the height of the cube is 'x' centimeters so I can then again use Pythagorean Theory to find the equation of the diagonal going through the cube.

So I set all sides of the cube equal to 'x' and solve for a diagonal of one of the sides of the cube in order to get a side length ( 'A' ) of the diagonal within the cube. Using Pythagorean I come to the equation of:

[itex]x^2+x^2=C^2[/itex]

[itex]2x^2=C^2[/itex]

[itex] \sqrt2x^2 = C [/itex]

After that I know that the height of the cube is 'x' so once again using Pythagorean I can derive an equation for the diagonal through the cube.

[itex](\sqrt2x^2)^2 + x^2 = D^2[/itex]

[itex]3x^2 = D^2[/itex]

[itex]\sqrt3x^2 = d[/itex]

Now this is where I become confused in order to find the rate of change at 1 cm. Do I then find the derivative of [itex]\sqrt3x^2 = D[/itex] through implicit differentiation?

I.E:

[itex]D = \sqrt3x^2[/itex]

[itex]D' = (3x)/\sqrt3x^2 (dLength/dt)[/itex]

... I'm rather lost on what to do from here. Any help would be greatly appreciated.

Last edited: