Question

Find the surface area of the torus generated by revolving the circle given by \(r=2\) about the line \(r=5 \sec \theta\)

Short answer

The surface area of the torus, under the assumption of a fixed \(sec\ \theta\), is \(160\pi^3 sec\ \theta\).

Step-by-step solution

Find the circumference of the circle

The formula to find the circumference of a circle is \(2\pi r\). Given that the radius is 2, then applying the formula we get \(2\pi*2 = 4\pi\).

Find the path circumference

The path circumference describes the path the center point of the circle follows when generating the torus. It is given by the formula \(2\pi d\), where \(d = 5 sec\ \theta\). The term \(sec\ \theta\) represents the hypotenuse divided by the adjacent side in trigonometry and varies with \(\theta\). However, assuming it as a fixed constant, the path circumference is \(2\pi d = 2\pi*5 sec\ \theta = 10\pi sec\ \theta\). Please note this value should be calculated at each point of rotation for accurate results.

Calculate the surface area of the torus

The formula for the surface area of the torus is the multiplication of the square of the circumference of the circle by the circumference of the path. Hence, the surface area is \((4\pi)^2*10\pi sec\ \theta = 160\pi^3 sec\ \theta\). The surface area varies with \(\theta\) and should be calculated at each point of rotation \(\theta\) for accurate results.