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, PQRS is a square of side 2 cm and PLMN is a rectangle. The corner L of the rectangle is on the side QR. Side MN of the rectangle passes through the corner S of the square.
What is the area (in cm²) of the rectangle PLMN?
Note: The figure shown is representative.
A regular dodecagon (12-sided regular polygon) is inscribed in a circle of radius $r$ cm as shown in the figure. The side of the dodecagon is $d$ cm. All the triangles (numbered 1 to 12) in the figure are used to form squares of side $r$ cm and each numbered triangle is used only once to form a square.
The number of squares that can be formed and the number of triangles required to form each square, respectively, are:
Note: The figure shown is representative.
- Which one of the following options has the correct sequence of objects arranged in the increasing number of mirror lines (lines of symmetry)?
- A circle with center at $(x, y) = (0.5, 0)$ and radius $= 0.5$ intersects with another circle with center at $(x, y) = (1, 1)$ and radius $= 1$ at two points. One of the points of intersection $(x, y)$ is:
- During a half-moon phase, the Earth-Moon-Sun form a right triangle. If the Moon-Earth-Sun angle at this half-moon phase is measured to be $89.85^{\circ}$, the ratio of the Earth-Sun and Earth-Moon distances is closest to
