More rates and changes help needed: (1 Viewer)

[cssftw]

Q. A block of ice in the form of a CUBE has one edge 10cm long. It is melting so that its dimensions decrease at the rate of 1mm/s (the block always remains a cube).

At what rate is the DIAGONAL decreasing:

(a) initially? (when edgelength = 10cm)?

(b) when the edge is 5cm long??


My solution:

Let edge length = x
Let diagonal = h
we know dx/dt = -1mm/s

dh/dt = dx/dt * dh/dx

to find dh/dx -- we need to find an expression of h in terms of x.

Using pythagoras on the square (side face of cube):

x^2 + x^2 = h^2
2x^2 = h^2
therefore h = sqrt(2)*x (h>0, x>0)

therefore dh/dx = sqrt(2)

therefore dh/dt = -1(sqrt(2))

dh/dt = -sqrt(2) mm/s --> which would imply that the rate is constant.

So is the answer -sqrt(2) mm/s i.e -0.141... cm/s??

Could someone please check the answer for me? thanks

 

[funnytomato]

-sqrt(2)= ??

 

[funnytomato]

the diagonal and the side are of the same dimension, it makes sense that they're both decreasing at constant rates to keep the shape as a square

 

[funnytomato]

e.g
after 9 seconds, the sides are 1 , and the diagonal is 10sqrt(2)- 9*sqrt(2)=sqrt(2)
which means it's still a square

 

[cssftw]

So is the actual answer -sqrt(2) mm/s?? Sorry but could you please check for me? I'm not exactly confident with my mathematical abilities...