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 by which the cone travels about point P until point Q touches the ground again, we need to understand the geometry involved.
The cone is essentially a sector of a circle when flattened. The length of the arc of this sector is equal to the circumference of the base of the cone.
The circumference of the base of the cone is given by:
- \(C = 2\pi r\)
where \(r = 5 \text{ cm}\).
- \(C = 2\pi \times 5 = 10\pi \text{ cm}\)
The length of the arc of the sector is also equal to the lateral height of the cone, which can be calculated using the Pythagorean theorem:
The slant height \(l\) is:
- \(l = \sqrt{r^2 + h^2}\)
where \(h = 12 \text{ cm}\)
- \(l = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \text{ cm}\)
The angle subtended by the arc at the center of the sector (say \(\theta\)) is proportional to the ratio of the arc length to the full length of the circle's circumference (if it were a full circle of the cone's lateral height).
This is given by:
- \(\theta = \frac{\text{Arc Length}}{\text{Circumference of the full circle}} \times 2\pi\)
Thus:
- \(\theta = \frac{10\pi}{13} \times 2\pi = \frac{10\pi}{13} \text{ radians}\)
This matches the given correct answer option \(\frac{10\pi}{13}\).
Therefore, the cone travels through an angle of \(\frac{10\pi}{13}\) radians about point P.
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.
