A right-angled cone (with base radius 5 cm and height 12 cm), as shown in the figure below, is rolled on the ground keeping the point P fixed until the point Q (at the base of the cone, as shown) touches the ground again. By what angle (in radians) about P does the cone travel?
To find the angle about point P by which the cone travels, we need to analyze the rolling motion of the cone.
Step-by-Step Solution:
- First, observe that the cone is rolled on the ground, keeping point P fixed, until point Q touches the ground again. This means the cone makes a complete rotation about point P.
- The slant height of the cone forms the radius of the sector when unfolded. To find the slant height, use the Pythagorean theorem:
\(l = \sqrt{r^2 + h^2} = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \, \text{cm}\)
- The circumference of the base of the cone is given by:
\(C = 2\pi r = 2\pi \times 5 = 10\pi \, \text{cm}\)
- When the cone is rolled, the arc length covered by the base of cone equals the circumference of the base.
- The angle of rotation about point P is given by the ratio of the arc length (circumference) to the circumference of the circle with radius \(l\) (when developed).
\(\text{Arc length} = 10\pi\)
\(\text{Circumference of circle with radius } l = 2\pi \times l = 2\pi \times 13 = 26\pi\)
The angle in radians is:
\(\theta = \frac{\text{Arc length}}{\text{circumference with radius} \, l} = \frac{10\pi}{26\pi} = \frac{10}{26} = \frac{5}{13}\)
Therefore, the angle by which the cone travels about point P is \(\frac{10\pi}{13}\) radians.
Thus, the correct answer is \(\frac{10\pi}{13}\).
Important Questions from Mensuration and Geometry
The city of Atlantis was crafted by the God of the seas, Poseidon. It was made of alternating concentric circular rings of land (shaded) and water (not shaded) as represented in the figure (not to scale). The radius of Inner Island was 2.5 stades (a unit of length used in ancient Greece). The water surrounding Inner Island was one stade wide (length AB). This was surrounded by two pairs of alternating rings of land and water. The first pair of land and water was two stades wide each (lengths BC and CD), and the outer pair is three stades wide each (lengths DE and EF).
The ratio of the surface area of the land to that of the water in the city of Atlantis is _________ (round off to two decimal places).
In the given figure, $P, Q$, and $R$ are three points on a circle of radius 10 cm with $O$ as its center, $\overline{PQ} = \overline{RQ}$, and $\angle PQR = 45^\circ$. The figure is representative.
The area of the shaded region $PQRO$ is ______________ cm$^2$.
- A straight line $y = x - 1$ intersects a circle with center at $x = 1, y = 1$ and radius of magnitude 1 at two points. The length of the chord formed by this intersection is _______. (rounded off to three decimal places)
- The shell of a hollow spherical nanoparticle has a uniform thickness of 3 nanometers (nm). The outer radius of the nanoparticle is 5 nm. The ratio of the volume of the shell to the volume of the hollow core is ________
(Round off to one decimal place) - The volume of a sphere of diameter 1 unit is ______ than the volume of a cube of side 1 unit.
