The volume of a cube is decreasing 10cm^3/min. How fast is the Surface Area decreasing when the length of one edge is 30cm?
If is the side of the cube then thus taking the derivative to respect to time from both time we have , conditions of the problem says the edge is 30 thus, and the rate of change is 10, thus, . Thus, placing in these values,Originally Posted by nirva
, thus,
. But this tells you the instantenous rate of change of the side, the problem asks for the instantenous rate of change of the surface area. Since for the cube then taking the derivate to respect to time we have,
know we know that for those are the conditions of this problem and in the other problem you found that thus, substituting these values,
.
Just placing in the proper unit values to get,
Here is one way.Originally Posted by nirva
Cube of side s cm.
Volume, V = s^3 -----------------------(1)
Surface area, A = 6(s^2) = 6s^2 -------(2)
From (1),
s = V^(1/3)
Substitute that into (2),
A = 6(V^(1/3))^2
A = 6V^(2/3)
Differentiate both sides with respect to time t,
dA/dt = 6(2/3)*V^(-1/3)*(dV/dt)
Plug into that the given dV/dt = -10 cu.cm/min,
dA/dt = 4*V^(-1/3) *(-10)
dA/dt = (-40)/[cubrt(V)] ---------------(3)
So when s=30cm,
V = (30)^3 cu.cm. Plug that into (3),
dA/dt = (-40)/[cubrt(30^3)]
dA/dt = -40/[30] = -4/3 sq.cm/min
Therefore, at that time, the surface area is decreasing by 4/3 sq.cm/min. ------------answer.
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Another way.
V = s^3
dV/dt = 3s^2 *(ds/dt) --------(i)
A = 6s^2
dA/dt = 12s*(ds/dt) ----------(ii)
When s=30cm, and that dV/dt is always -10cu.cm/min, then, plug those into (i),
-10 = 3(30^2)*(ds/dt)
-10 = 2700(ds/dt)
ds/dt = -10/2700 = -1/270 cm/min.
Substitute that, and s=30cm, into (ii),
dA/dt = 12(30)(-1/270)
dA/dt = -4(10)(1/30)
dA/dt = -40/30 = -4/3 sq.cm/min. ---------same as above.
Hello,Originally Posted by nirva
here is a more practical way: There are 4 squares involved, when the surface area decreases. I've attached a drawing to demonstrate, what I would like to explain.
So the part of the surface, which is "lost" per minute, is a rectangle with 4*30 cm length and height. That means, it's an area of per min.
Greetings
EB