To find the volume of the oxygen tank, we need to combine the volumes of the cylindrical part and the two hemispheres at the top and bottom.

1. Cylinder Volume:
   The volume of a right circular cylinder is given by the formula: $V_{\text{cylinder}} = \pi x^2 y$ where $x$ is the radius of the base, and $y$ is the height of the cylinder.

2. Two Hemispheres Volume:
   The volume of a single sphere is: $V_{\text{sphere}} = \frac{4}{3} \pi x^3$ Since there are two hemispheres (which together make a full sphere), their combined volume is the same as the volume of one sphere: $V_{\text{hemispheres}} = \frac{4}{3} \pi x^3$

3. Total Volume of the Tank:
   The total volume $V(x, y)$ of the oxygen tank is the sum of the volumes of the cylindrical part and the hemispheres: $V(x, y) = \pi x^2 y + \frac{4}{3} \pi x^3$

Thus, the formula for the total volume is: $V(x, y) = \pi x^2 y + \frac{4}{3} \pi x^3$

Finding $V(11, 3)$

Now, substitute $x = 11$ and $y = 3$ into the formula: $V(11, 3) = \pi (11)^2 (3) + \frac{4}{3} \pi (11)^3$

First, compute the two parts: $\pi (11)^2 (3) = \pi (121)(3) = 363\pi$ $\frac{4}{3} \pi (11)^3 = \frac{4}{3} \pi (1331) = \frac{5324}{3} \pi$

Now, sum these up: $V(11, 3) = 363\pi + \frac{5324}{3}\pi = \frac{1089}{3}\pi + \frac{5324}{3}\pi = \frac{6413}{3}\pi$

Approximating this using $\pi \approx 3.1416$: $V(11, 3) \approx \frac{6413}{3} \times 3.1416 = 6711.37 \, \text{units}^3$

Explanation of the result:

This means that an oxygen tank with a cylindrical height of 3 units and a radius of 11 units has a total volume of approximately 6711.37 cubic units.

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Do you want more details or have any questions?

Here are 5 related questions:

1. How does the volume formula change if the hemispheres are replaced with cones?
2. What is the surface area of this oxygen tank?
3. How would you express the volume if only one hemisphere is attached to the cylinder?
4. What is the total height of the tank, including both hemispheres?
5. How would the volume change if the height of the cylinder is doubled?

Tip: Understanding how to split shapes into simpler components can make solving geometric problems much easier.