Optimization Problem!

Jun 2010
2
0
The radius and height of a tank cannot exceed 5 m and 12 m respectively. Is it possible to construct a tank with a volume of 900 m3? If it is possible, what are the restrictions on its radius?

Can anyone help me?

Thanks!
 
May 2010
1,034
272
you didn't say what shape the tank is, and i dont think this is a calculus problem. Are we to assume its a cylinder?

if so, you know the formula for the volume of a cylinder \(\displaystyle V= \pi r^2 h\)

You can easily use that to check is it's possible to construct a tank of the required size.


To see find the minimum radius (for a tank of size 900), look for the radius that corresponds to the maximum height.
 
Feb 2010
168
20
Columbus, Ohio, USA
The radius and height of a tank cannot exceed 5 m and 12 m respectively. Is it possible to construct a tank with a volume of 900 m3? If it is possible, what are the restrictions on its radius?

Can anyone help me?

Thanks!
I am going to assume that when you said "tank" you were refering to a cylynder (mispelling) of some sorts. So lets call the radius \(\displaystyle r\) and the hiegth \(\displaystyle h\). By the initial conditions given in the first sentence we have the following relations:

\(\displaystyle h \leq 5\) and \(\displaystyle r \leq 12\)

Since both h and r are lengths, and we are using them as such, we know that:

\(\displaystyle h \geq 0\) and \(\displaystyle r \geq 0\)

From this it follows that:

\(\displaystyle 0 \leq h \leq 5\) and \(\displaystyle 0 \leq r \leq 12\)

For the area, we have the following:

\(\displaystyle A = (h)(\pi)(r^2)\)

This can be viewed as a function of a surface of the form \(\displaystyle f(h, r) = A\), with the domains of the function defined above with the inequalities. All you need to do is determine weather \(\displaystyle f(h, r)\) ever obtains the value of 900, and then determine at which values of \(\displaystyle h\) and \(\displaystyle r\) [corresponding to \(\displaystyle (x, y)\) on a graph] at which the 900 occurs, and you must make sure to be looking only in the specified domains.
 
Jun 2010
2
0
Sorry forgot to mention. It is a cylinder.