How to use differentiation to determine the value of x for which A is a minimum.?

Question

A cuboid of height $h$ cm with a rectangular base that has sides of length $x$ cm and $2x$ cm. Its volume $V$ is required to be $1000$ cm$^3$.

a) Write down an expression for the volume $V$ in terms of $x$ and $h$. Hence find an expression for $h$ in terms of $x$.

b) Write down an expression for the surface area $A$ of the box in terms of $h$ and $x$, and use the result from part (a) to write $A$ in terms of $x$ only.

c) Use differentiation to determine the value of $x$ for which $A$ is a minimum. What is the corresponding value of $h$?

Ok so I have

a) (I think) $V=2x^2h\Rightarrow h=\dfrac{V}{2x^2}$

b) $A=2(2x^2+x\cdot h+2x\cdot h)$

How do I write $A$ in terms of $x$ only? and question c, I really do not understand. I am finding that differentiation is my nemesis!

Answer 1

note that we have $$1000cm^{3}=2x^2\cdot h$$ can you go on? from here we get $$h=\frac{1000}{2x^2}$$ plugging this in your formula we obtain $$A=2(2x^2+3x\frac{1000}{2x^2})$$ this is a function only in $x$ it simplifies to $$A=2(2x^2+\frac{150}{x})$$ and $$A'(x)=2(4x-\frac{150}{x^2})$$

Answer 2

From your answer to a) you get that $h=\frac{1000}{2x^2}=\frac{500}{x^2}$. From this and your answer to b) you get that$$A=2\left(2x^2+\frac{500x}{x^2}+\frac{1000x}{x^2}\right)=4x^2+\frac{3000}x.$$Now, can you use differentiation to determine the minimum of $A$?