# Thread: [SOLVED] Applications of Differentiation Question

1. ## [SOLVED] Applications of Differentiation Question

Question:
A hollow circular cylinder, open at one end, is constructed of thin sheet metal. The total external surface area of the cylinder is $\displaystyle 192\pi$$\displaystyle cm^2. The cylinder has a radius of \displaystyle r$$\displaystyle cm$ and a height of $\displaystyle h$$\displaystyle cm. (i) Express h in terms of r and show that the volume, \displaystyle V$$\displaystyle cm^3$, of the cylinder is given by:
$\displaystyle V = \frac{1}{2}\pi(192r - r^3)$

Given that $\displaystyle r$ can vary,
(ii) Find the value of $\displaystyle r$ for which $\displaystyle V$ has a stationary value,
(iii) find this stationary value and determine whether it is a maximum or a minimum.

(i) Can someone please show me how to get $\displaystyle V = \frac{1}{2}\pi(192r - r^3)$?

2. external surface area of a closed cylinder
= $\displaystyle 2\pi r h + 2\pi r^2$

external surface area for one open end of cylinder
= $\displaystyle 2\pi r h + \pi r^2$
= $\displaystyle 192\pi$

obtain an expression for h.

volume of cylinder
= $\displaystyle \pi r^2 h$

substitute the value of h into the volume equation

external surface area of a closed cylinder
= $\displaystyle 2\pi r h + 2\pi r^2$

external surface area for one open end of cylinder
= $\displaystyle 2\pi r h + \pi r^2$
= $\displaystyle 192\pi$

obtain an expression for h.

volume of cylinder
= $\displaystyle \pi r^2 h$

substitute the value of h into the volume equation

$\displaystyle 2\pi{rh} + \pi{r^2} = 192\pi$

$\displaystyle \frac{2\pi{rh}}{\pi} + \frac{\pi{r^2}}{\pi} = \frac{192\pi}{\pi}$

$\displaystyle 2rh + r^2 = 192$

$\displaystyle 2rh = 192 - r^2$

$\displaystyle h = \frac{192 - r^3}{2r}$

$\displaystyle = \pi{r^2h}$

$\displaystyle =\pi{r^2}\frac{192 - r^3}{2r}$

$\displaystyle V = \frac{1}{2}\pi{}(192r - r^3)$

Now how do I find the value of r?

4. Definition:A point 'u' at which the derivative of a function f(x) vanishes,i.e. f'(u) = 0 is called a stationary point.

This means you have to find values of r such that V'(r) = 0.

5. yup just like what isomorphism said, a stationary point can be found when the derivative of V with respect to r is 0

$\displaystyle \frac{dV}{dr} = 0$

earlier you already prove the equation of the volume:

$\displaystyle V = \frac{1}{2}\pi{}(192r - r^3)$

hence all you need to do is to differentiate this equation with respect to r and equate it to 0:

$\displaystyle \frac{d}{dr}{(\frac{1}{2}\pi{}(192r - r^3)}) = 0$

solve for r. good luck

,
,

,

,

,

,

,

,

,

,

,

,

,

,

# A circular cylinder open at one end is constructed of thin sheet metal whose area is 432pi cmÂ².The cylinder has a radiua r cm and a height h cm. Show that the volume V cm contained by the cylinder is given by V=pi/2 (432r - rÂ²)

Click on a term to search for related topics.