Related Rates - Cube

**sparky** · February 28th, 2011, 07:43 PM

Hi,

I need some guidance on this question please:

Question: The volume of a cube is increasing at $10cm^3/min$ . At the instant when its volume is $125 cm^3$ , determine the rate of change of the edge and its total surface area.

I am confused about how to determine the rate of change of the edge and what exactly is the edge on the cube, but here is what I have so far:

$V = x^3$

$\frac{dV}{dt}= 3x^2 \cdot \frac{dx}{dt}$

$125 = 3x^2 \cdot \frac{dx}{dt}$

$\frac{125}{3x^2}=\frac{dx}{dt}$ (I am lost at this point)

**pickslides** · February 28th, 2011, 07:49 PM

I get $\displaystyle \frac{dV}{dt}= 10 \neq 125$

**sparky** · February 28th, 2011, 08:20 PM

Thanks for the reply.

$V = x^3$

$\frac{dV}{dt}= 3x^2 \cdot \frac{dx}{dt}$

$10 = 3x^2 \cdot \frac{dx}{dt}$

$\frac{10}{3x^2}=\frac{dx}{dt}$

I am not sure how to proceed from here...

**NOX Andrew** · February 28th, 2011, 08:22 PM

To find x, solve the equation $V = x^3$ (where $V = 125$ ). Then, substitute the value of x into the other equation to find $\dfrac{dx}{dt}$ .

**sparky** · March 1st, 2011, 05:56 PM

Ok NOX, is my working and logic correct now?

$V = x^3$

$\frac{dV}{dt}= 3x^2 \cdot \frac{dx}{dt}$

$10 = 3x^2 \cdot \frac{dx}{dt}$

We want to find out the value of $\frac{dx}{dt}$ (which is the rate of change of the edge), therefore we need to find the value of x first

$V = x^3$ ,

Therefore $125 = x^3$

$\sqrt[3]{125} = x$

$x = 5$

Therefore, $10 = 3(5)^2 \cdot \frac{dx}{dt}$

$10 = 75 \cdot \frac{dx}{dt}$

$\frac{10}{75}=\frac{dx}{dt}$

$\frac{dx}{dt}=0.133cm/s$ (rate of change of the edge)

Total surface area $= x \cdot x \cdot 6$

$= 5 \cdot 5 \cdot 6$

$= 150 cm^3$

**skeeter** · March 1st, 2011, 06:06 PM

I think the question is asking for the rate of change of the surface area, not the actual surface area (which would have units of cm squared rather than cubed) ...

**NOX Andrew** · March 1st, 2011, 06:23 PM

The units of surface area should be centimeters squared (5 cm * 5 cm * 6). Otherwise, your logic is flawless.

**sparky** · March 1st, 2011, 10:32 PM

$= 150 cm^2$

Thanks

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