# Optimization Problem!

#### ofmic3

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!

#### SpringFan25

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.

#### mfetch22

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.

#### ofmic3

Sorry forgot to mention. It is a cylinder.