Circles (arc length, sector area): notes and practice questions

A graphic designer is creating a logo for a new technology company. The logo features a stylized 'C' shape, which is a circular arc. This arc is part of a circle with centre O and a radius of 8 cm. The arc subtends an angle of 120° at the centre O. The straight edge of the 'C' is a chord connecting the endpoints of the arc.

The logo component is shown as the curved boundary in the following diagram.

(a) Find the length of the curved boundary of this logo component.

(b) Find the area of the region enclosed by the arc and the straight edge (the segment of the circle).

A large water wheel is used for irrigation. The lowest point a bucket on the wheel reaches is $0.5$ m above the water surface, and its highest point is $12.5$ m above the water surface.

(a) (i) Show that the radius of the water wheel is $6$ m.

(ii) Calculate the circumference of the water wheel.

(b) When the wheel rotates $15^\circ$, find the distance that a bucket travels along the circumference.

The height in metres, above the water surface, of a particular bucket is modelled by the function:

$h(t) = a \sin (bt) + d$, for $a, b > 0$,

where $t$ is the time, measured in minutes.

The wheel takes $12$ minutes to complete $1$ revolution.

(c) (i) Find the value of $b$.

(ii) Find the value of $d$.

(iii) Hence, write down the equation of the sinusoidal model.

(d) Use the model to find the values of $t$ when the height of this bucket is $9$ m above the water surface for $0 \le t \le 12$.

The water wheel operates for $2500$ days, and each day it rotates nonstop for $10$ hours.

(e) Calculate the total number of rotations that the water wheel has made. Give your answer in the form $a \times 10^k$ where $1 \le a < 10$ and $k$ is an integer.

A landscape architect is designing a modular planter box for a new urban garden. The planter box has a base and top that are identical sectors of a circle, each with radius $r$ cm and angle $\theta$ radians. The height of the planter box is $h = 2$ cm. The total length of metal frame used for all edges of the planter box is $L = 20$ cm. This includes two circular arcs, four radial edges for the top and bottom sectors, and three vertical connecting edges.

(a) Show that $r = \frac{7}{2+\theta}$.

(b) The planter box is designed to hold soil, enclosing a volume, $V$.

(i) Find an expression for $V$ in terms of $\theta$.

(ii) Find the expression for $\frac{dV}{d\theta}$.

(iii) Solve algebraically $\frac{dV}{d\theta} = 0$ to find the value of $\theta$ that will maximize the volume, $V$.

(a) A ground crew on Luna Prime is tracking a supply drone. From the Aether station's control tower, the drone is observed to be $8500$ km East and $4000$ km North in a localized flat-map approximation.

(i) Find the straight-line distance from the Aether station to the drone.

(ii) Find the bearing of the drone from the Aether station. Give your answer in degrees, correct to one decimal place.

(b) The Aether station (A) is located at the origin $(0,0,0)$ of a 3D Cartesian coordinate system for this part. A research probe (P) is located at $(4000, 0, 0)$ km. A navigation beacon (B) is located at $(0, 4000, 0)$ km.

(i) Show that the position vector of the research probe, $\vec{p}$, is perpendicular to the position vector of the navigation beacon, $\vec{b}$.

(ii) Assuming the probe and beacon are on the surface of Luna Prime with a radius of $4000$ km, calculate the shortest distance along the surface between the research probe and the navigation beacon.

(c) Consider the Aether station (A) at $(4000, 0, 0)$ km, the Boreas station (B) at $(0, 4000, 0)$ km, and a North Pole reference point (N) at $(0, 0, 4000)$ km. Let $\vec{a}$, $\vec{b}$, and $\vec{n}$ be their respective position vectors from the centre of Luna Prime.

(i) Find the vector $\vec{a} \times \vec{b}$.

(ii) Show that the angle at vertex A in the spherical triangle formed by Aether, Boreas, and the North Pole (ABN) is $90^{\circ}$.

(d) A supply route between Aether and a new outpost, Delta, has an arc length of $2000$ km. Given that the radius of Luna Prime is $4000$ km, show that the central angle $\theta$ between Aether and Delta is $28.6^{\circ}$, correct to three significant figures.

(e) The Aether station (A) is located at $40^{\circ}$ N, $20^{\circ}$ E, and the Boreas station (B) is located at $70^{\circ}$ N, $100^{\circ}$ E on Luna Prime, which has a radius of $6000$ km. Find the shortest distance along the surface from Aether to Boreas.

(f) Using the vector method from part (c), find the initial bearing from Aether to Boreas. Give your answer in degrees, correct to one decimal place.

A robotic arm is designed to sweep through a fixed angle of 60 degrees. The arm starts with an initial length of 5 cm and then extends by 2 cm for each subsequent sweep. The path traced by the tip of the arm forms an arc of a circle.

(a) Find the length of the arc traced during the first sweep.

(b) Find the length of the arc traced during the second sweep.

(c) Find the length of the arc traced during the third sweep.

(d) Describe the relationship between the lengths of the arcs traced during the first, second, and third sweeps. Hence, find the lengths of the arcs traced during the fourth and fifth sweeps without performing full calculations.

(e) Determine the area of the sector traced by the arm during its sixth sweep.

Three emergency service stations are located at points $A(2, 1)$, $B(10, 5)$, and $C(4, 9)$ in a city's coordinate grid, where coordinates are in kilometres.

A central command hub is to be built at a point $T$, equidistant from all three stations.

(a)(i) Find the midpoint of the line segment $AB$.

(a)(ii) Find the gradient of the line $AB$.

(a)(iii) Hence, find the equation of the perpendicular bisector of the line segment $AB$.

(b)(i) Find the midpoint of the line segment $AC$.

(b)(ii) Find the gradient of the line $AC$.

(b)(iii) Hence, find the equation of the perpendicular bisector of the line segment $AC$.

(c) Determine the coordinates of the central command hub $T$.

(d) The city regulations state that the central command hub must be within a service radius of $5$ km from each station for effective communication. Determine if the proposed location $T$ meets this regulation.

(e) Calculate the area of the circular service region covered by the central command hub, assuming its effective radius is the distance to any of the stations. Give your answer to one decimal place.

A company is designing a new logo. Part of the logo involves a shaded region on a coordinate plane. This region is bounded by the line segment connecting the points $A(1, 4)$ and $B(5, 2)$, the $x$-axis, and the vertical lines $x=1$ and $x=5$.

Write down a definite integral that represents the area of this shaded region.

Calculate the area of the region.

Another part of the logo design involves a curved section. This section is represented by the region bounded by the curve $y = \sqrt{25 - x^2}$ and the $x$-axis.

Write down a definite integral that represents the area of this region.

Calculate the area of the region.

A circular park has a central monument. A straight walking path $AB$ cuts across the park. The radius of the park is $6$ m. The shortest distance from the center of the park $O$ to the path $AB$ is $5$ m.

(a) (i) Calculate the angle $A\hat{O}B$ in degrees.

(a) (ii) The region of the park bounded by the path $AB$ and the arc $AB$ is a flower bed. Calculate the area of this flower bed.

A square performance stage has a special effect light at its center. The stage has a side length of $10$ m. The light projects a circular pattern on the stage floor with a radius of $6$ m.

(b) (i) Calculate the total area of the circular light pattern if it were projected onto an infinite surface.

(b) (ii) Calculate the area of the stage floor that is lit by the special effect light.

Let $B$ be the brightness of the special effect light, measured in lumens, and $t$ be the time in minutes since the light was switched on.

The rate of change of brightness is given by $\frac{dB}{dt} = 0.5 t e^{-0.2t}$.

(c) Find the value of $t$ at which the brightness of the light is increasing at the greatest rate.

A drone is used for aerial surveillance of a research facility. Three key locations are the Launch Pad (L), a Data Collection Point (D), and a Scientist's observation post (S).

The distance between the Launch Pad (L) and the Data Collection Point (D) is $LD = 10.0$ m.

The Scientist (S) is observing such that $L\hat{S}D = 30^\circ$. The distance from the Scientist to the Launch Pad is $SL = 10.0$ m. The angle $S\hat{L}D$ is obtuse.

Calculate the size of $S\hat{D}L$.

Calculate the area of triangle $SDL$.

An Automated Repair Station (R) is located such that it is $LR = 8.0$ m from the Launch Pad (L). The angle $D\hat{L}R = 60^\circ$.

Calculate the distance between the Data Collection Point (D) and the Repair Station (R).

A new Target (T) is identified. The distance from the Repair Station (R) to the Target (T) is $RT = 22.2$ m. From the Scientist's observation post (S), the angle $R\hat{S}T = 53.8^\circ$. From the Repair Station (R), the angle $S\hat{R}T = 51.1^\circ$.

Determine whether the Scientist (S) or the Repair Station (R) is closer to the Target (T).

The drone needs to travel from the Repair Station (R) to the Target (T) along a semi-circular maintenance track. This track has its centre at the Launch Pad (L). The angle subtended by the arc $RT$ at the centre L is $110^\circ$.

Calculate the distance the drone travels along this track.