圓定理（高階）：練習題

A rectangular field has a length of 150 m and a width of 80 m.
Calculate the area of the field and give your answer in hectares.
(Note: \(1\text{ hectare} = 10,000 \text{ m}^2\))

In triangle \( ABC \), side \( AB = 7 \text{ cm} \), side \( AC = 10 \text{ cm} \) and angle \( BAC = 150^\circ \).
Calculate the area of the triangle.

The diagram shows a frustum of a cone. The original cone had a height of 20 cm and a base radius of 8 cm. A smaller cone of height 5 cm was removed from the top.
Calculate the volume of the frustum in terms of \(\pi\).

Write down the exact value of \(\cos(60^\circ)\).

The diagram shows a cylinder with a radius of 3 cm and a height of 10 cm.
Calculate the volume of the cylinder. Leave your answer in terms of \(\pi\).

Write down the mathematical name for a polygon with five sides.

In a right-angled triangle, the two shorter sides have lengths of 6 cm and 8 cm.
Calculate the length of the hypotenuse.

A solid sphere has a radius of 5 cm.
Calculate the total surface area of the sphere, giving your answer in terms of \(\pi\).
(The formula for the surface area of a sphere is \(4\pi r^2\)).

In the diagram, \(A\), \(B\), and \(C\) are points on a circle with centre \(O\). \(PAT\) is a tangent to the circle at \(A\).
Angle \(BAP = 58^\circ\) and angle \(ABC = 72^\circ\).

(a) Find the size of angle \(ACB\). Give a reason for your answer.
(b) Find the size of angle \(AOC\).
(c) Given the radius of the circle is \(8\text{ cm}\), calculate the area of the minor sector \(AOC\). Give your answer to 1 decimal place.

Two ships, A and B, leave a port P at the same time.
Ship A travels on a bearing of \( 040^\circ \) at a constant speed of \( 15 \text{ km/h} \).
Ship B travels on a bearing of \( 130^\circ \) at a constant speed of \( 22 \text{ km/h} \).

(a) Calculate the distance between ship A and ship B after 2 hours. Give your answer to 1 decimal place.
(b) Find the bearing of ship A from ship B at this time. Give your answer to the nearest degree.