Question

A propane tank is constructed by welding hemispheres to the ends of a right circular cylinder. Write the volume \(V\) of the tank as a function of \(r\) and \(l,\) where \(r\) is the radius of the cylinder and hemispheres, and \(l\) is the length of the cylinder.

Short answer

The volume \(V\) of the propane tank as a function of \(r\) and \(l\) is \(V = \pi r^2 l + \frac{4}{3}\pi r^3\).

Step-by-step solution

Volume Calculation of the Cylinder

The volume of a cylinder is given by the formula \(\pi r^2 h,\) where \(r\) is the radius and \(h\) is the height of the cylinder. Here, the height of the cylinder is same as length \(l\). So the volume \(V_c\) of the cylinder can be calculated using \(\pi r^2 l\).

Volume Calculation of the Hemisphere

The volume of a hemisphere is given by the formula \(\frac{2}{3}\pi r^3,\), where \(r\) is the radius of the hemisphere. The propane tank has two hemispheres so the combined volume \(V_h\) of the hemispheres is \(2*(\frac{2}{3}\pi r^3)\) which simplifies to \(\frac{4}{3}\pi r^3\).

Total Volume Calculation of the Propane Tank

Now, the volume \(V\) of the propane tank is the sum of the volume of the cylinder \(V_c\) and volume of the two hemispheres \(V_h\). Hence, substituting the respected formulas, we get \(V = \pi r^2 l + \frac{4}{3}\pi r^3\).