Calculating Paint Usage on a Cube with Differentials

[Martin Spasov]

Homework Statement

A coat of paint of thickness 0.02 in is applied to the faces of a cube whose edge is 10 in, thereby producing a slightly larger cube. Use differentials to find approximately the number of cubic inches of paint used. Also find the exact amount used by computing volumes before and after painting.

Homework Equations

V = x³

f'(x) = 3x²

dy = f'(x)dx

The Attempt at a Solution

dy = 3*10²*0.02 = 6

However the actual solution is 12 (from the answers). Even when doing it manually :

6 * 10² * 0.02 = 12

When I compute the volume before and after I get the same result as before :

10.02³ - 10³ = 6.012008

I can clearly see that there is factor of 2 difference, but why ? I used the formula and did not get the solution, what exactly was not ok ?

p.s. first post, if I did something wrong, please point it out :)

 

[Ray Vickson]

Homework Statement

A coat of paint of thickness 0.02 in is applied to the faces of a cube whose edge is 10 in, thereby producing a slightly larger cube. Use differentials to find approximately the number of cubic inches of paint used. Also find the exact amount used by computing volumes before and after painting.

Homework Equations

V = x³

f'(x) = 3x²

dy = f'(x)dx

The Attempt at a Solution

dy = 3*10²*0.02 = 6

However the actual solution is 12 (from the answers). Even when doing it manually :

6 * 10² * 0.02 = 12

When I compute the volume before and after I get the same result as before :

10.02³ - 10³ = 6.012008

I can clearly see that there is factor of 2 difference, but why ? I used the formula and did not get the solution, what exactly was not ok ?

p.s. first post, if I did something wrong, please point it out :)

For sides of length $x$ the total surface area is $A = 6 x^2$, because there are 6 faces of area $x^2$ each. What is $dA$?

 

[fresh_42]

By which amount of $dx$ did you increase an edge?

 

[Ray Vickson]

By which amount of $dx$ did you increase an edge?

I'm leaving that up to the OP to think about.

 

[fresh_42]

I'm leaving that up to the OP to think about.

That's how it was meant. You beat me while I was typing it. But have you noticed that the OP now has solutions in 1,2 and 3 dimensions? Sorry, someone deleted the 3-dimensional one.
Edit: What a pitty. It was a funny situation.

 

[Martin Spasov]

Thanks guys, the problem was hat i was equating dx to 0.02 and not to 0.04. Now everything worked out fine :)