How do i solve this:
A rectangular block of metal with a square cross-section has a total surface area of 625cm2. Find the maximum volume of the block of metal?
Help!!
Let the rectangular block have dimensions x, x and y. (2 x's because the cross section is a square).
So the surface area is given be $\displaystyle 2x^2$ (each end face) $\displaystyle + 4xy$ (each length face).
$\displaystyle 2x^2 + 4xy = 625$
$\displaystyle y = \frac{625 - 2x^2}{4x}$
Now,
Volume = 'area of cross section' times 'length' = $\displaystyle x^2y$.
$\displaystyle = x^2 \times \frac{625 - 2x^2}{4x}$
$\displaystyle = x \times \frac{625 - 2x^2}{4}$
$\displaystyle = \frac{1}{4} \left({625x - 2x^3}\right)$
$\displaystyle \frac{dV}{dx} = \frac{1}{4} (625 - 6x^2) = 0$ (for max)
$\displaystyle 6x^2 = 625$
$\displaystyle x = \sqrt{\frac{625}{6}}$
Once you have calculated this value, substitute into the equation for y. Use the Volume = $\displaystyle x^2y$ formula to find the max volume.