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
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.
- A window is made up of a square portion and an equilateral triangle portion above it. The base of the triangular portion coincides with the upper side of the square. If the perimeter of the window is 6 m, the area of the window in $m^2$ is ______
