1.05d Radians: arc length s=r*theta and sector area A=1/2 r^2 theta

8. \begin{figure}[h] \end{figure} The compound shape \(A B C D A\), shown in Figure 1, consists of a triangle \(A B D\) joined along its edge \(B D\) to a sector \(D B C\) of a circle with centre \(B\) and radius 6 cm . The points \(A , B\) and \(C\) lie on a straight line with \(A B = 5 \mathrm {~cm}\) and \(B C = 6 \mathrm {~cm}\). Angle \(D A B = 1.1\) radians.

1. Show that angle \(A B D = 1.20\) radians to 3 significant figures.
2. Find the area of the compound shape, giving your answer to 3 significant figures.

8. \begin{figure}[h] \end{figure} Figure 3 shows a circle with centre \(O\) and radius \(r \mathrm {~cm}\). The points \(A\) and \(B\) lie on the circumference of this circle. The minor arc \(A B\) subtends an angle \(\theta\) radians at \(O\), as shown in Figure 3.
Given the length of minor \(\operatorname { arc } A B\) is 6 cm and the area of minor sector \(O A B\) is \(20 \mathrm {~cm} ^ { 2 }\),

1. write down two different equations in \(r\) and \(\theta\).
2. Hence find the value of \(r\) and the value of \(\theta\).
   

15. \begin{figure}[h] \end{figure} Figure 2 shows a plan for a garden.
The garden consists of two identical rectangles of width \(y \mathrm {~m}\) and length \(x \mathrm {~m}\), joined to a sector of a circle with radius \(x \mathrm {~m}\) and angle 0.8 radians, as shown in Figure 2. The area of the garden is \(60 \mathrm {~m} ^ { 2 }\).

1. Show that the perimeter, \(P \mathrm {~m}\), of the garden is given by $$P = 2 x + \frac { 120 } { x }$$
2. Use calculus to find the exact minimum value for \(P\), giving your answer in the form \(a \sqrt { b }\), where \(a\) and \(b\) are integers.
3. Justify that the value of \(P\) found in part (b) is the minimum. \includegraphics[max width=\textwidth, alt={}, center]{1f61f78b-5e77-4758-8ad5-ea00c7dfea2b-49_83_59_2636_1886}

15. \begin{figure}[h] \end{figure} The triangle \(X Y Z\) in Figure 4 has \(X Y = 6 \mathrm {~cm} , Y Z = 9 \mathrm {~cm} , Z X = 4 \mathrm {~cm}\) and angle \(Z X Y = \alpha\). The point \(W\) lies on the line \(X Y\). The circular arc \(Z W\), in Figure 4 is a major arc of the circle with centre \(X\) and radius 4 cm .

1. Show that, to 3 significant figures, \(\alpha = 2.22\) radians.
2. Find the area, in \(\mathrm { cm } ^ { 2 }\), of the major sector \(X Z W X\). The region enclosed by the major arc \(Z W\) of the circle and the lines \(W Y\) and \(Y Z\) is shown shaded in Figure 4. Calculate
3. the area of this shaded region,
4. the perimeter \(Z W Y Z\) of this shaded region. \includegraphics[max width=\textwidth, alt={}, center]{1528bec3-7a7a-42c5-bac2-756ff3493818-39_90_54_2576_1868}

7. \begin{figure}[h] \end{figure} Figure 1 shows the triangle \(A B C\), with \(A B = 8 \mathrm {~cm} , A C = 11 \mathrm {~cm}\) and \(\angle B A C = 0.7\) radians. The \(\operatorname { arc } B D\), where \(D\) lies on \(A C\), is an arc of a circle with centre \(A\) and radius 8 cm . The region \(R\), shown shaded in Figure 1, is bounded by the straight lines \(B C\) and \(C D\) and the \(\operatorname { arc } B D\). Find

1. the length of the \(\operatorname { arc } B D\),
2. the perimeter of \(R\), giving your answer to 3 significant figures,
3. the area of \(R\), giving your answer to 3 significant figures.

5. \begin{figure}[h] \end{figure} In Figure \(2 O A B\) is a sector of a circle radius 5 m . The chord \(A B\) is 6 m long.

1. Show that \(\cos A \hat { O } B = \frac { 7 } { 25 }\).
2. Hence find the angle \(A \hat { O } B\) in radians, giving your answer to 3 decimal places.
3. Calculate the area of the sector \(O A B\).
4. Hence calculate the shaded area.

9. \begin{figure}[h] \end{figure} Figure 2 shows a plan of a patio. The patio \(P Q R S\) is in the shape of a sector of a circle with centre \(Q\) and radius 6 m . Given that the length of the straight line \(P R\) is \(6 \sqrt { } 3 \mathrm {~m}\),

1. find the exact size of angle \(P Q R\) in radians.
2. Show that the area of the patio \(P Q R S\) is \(12 \pi \mathrm {~m} ^ { 2 }\).
3. Find the exact area of the triangle \(P Q R\).
4. Find, in \(\mathrm { m } ^ { 2 }\) to 1 decimal place, the area of the segment \(P R S\).
5. Find, in \(m\) to 1 decimal place, the perimeter of the patio \(P Q R S\).

7. \begin{figure}[h] \end{figure} The shape \(B C D\) shown in Figure 3 is a design for a logo. The straight lines \(D B\) and \(D C\) are equal in length. The curve \(B C\) is an arc of a circle with centre \(A\) and radius 6 cm . The size of \(\angle B A C\) is 2.2 radians and \(A D = 4 \mathrm {~cm}\). Find

1. the area of the sector \(B A C\), in \(\mathrm { cm } ^ { 2 }\),
2. the size of \(\angle D A C\), in radians to 3 significant figures,
3. the complete area of the logo design, to the nearest \(\mathrm { cm } ^ { 2 }\).

4. \begin{figure}[h] \end{figure} An emblem, as shown in Figure 1, consists of a triangle \(A B C\) joined to a sector \(C B D\) of a circle with radius 4 cm and centre \(B\). The points \(A , B\) and \(D\) lie on a straight line with \(A B = 5 \mathrm {~cm}\) and \(B D = 4 \mathrm {~cm}\). Angle \(B A C = 0.6\) radians and \(A C\) is the longest side of the triangle \(A B C\).

1. Show that angle \(A B C = 1.76\) radians, correct to 3 significant figures.
2. Find the area of the emblem.

7. \begin{figure}[h] \end{figure} Figure 2 shows \(A B C\), a sector of a circle of radius 6 cm with centre \(A\). Given that the size of angle \(B A C\) is 0.95 radians, find

1. the length of the \(\operatorname { arc } B C\),
2. the area of the sector \(A B C\). The point \(D\) lies on the line \(A C\) and is such that \(A D = B D\). The region \(R\), shown shaded in Figure 2, is bounded by the lines \(C D , D B\) and the \(\operatorname { arc } B C\).
3. Show that the length of \(A D\) is 5.16 cm to 3 significant figures. Find
4. the perimeter of \(R\),
5. the area of \(R\), giving your answer to 2 significant figures.

7. \begin{figure}[h] \end{figure} The triangle \(X Y Z\) in Figure 1 has \(X Y = 6 \mathrm {~cm} , Y Z = 9 \mathrm {~cm} , Z X = 4 \mathrm {~cm}\) and angle \(Z X Y = \alpha\). The point \(W\) lies on the line \(X Y\). The circular arc \(Z W\), in Figure 1 is a major arc of the circle with centre \(X\) and radius 4 cm .

1. Show that, to 3 significant figures, \(\alpha = 2.22\) radians.
2. Find the area, in \(\mathrm { cm } ^ { 2 }\), of the major sector \(X Z W X\). The region enclosed by the major arc \(Z W\) of the circle and the lines \(W Y\) and \(Y Z\) is shown shaded in Figure 1. Calculate
3. the area of this shaded region,
4. the perimeter \(Z W Y Z\) of this shaded region.

8. \begin{figure}[h] \end{figure} Figure 2 shows a circle \(C\) with centre \(O\) and radius 5

1. Write down the cartesian equation of \(C\). The points \(P ( - 3 , - 4 )\) and \(Q ( 3 , - 4 )\) lie on \(C\).
2. Show that the tangent to \(C\) at the point \(Q\) has equation $$3 x - 4 y = 25$$
3. Show that, to 3 decimal places, angle \(P O Q\) is 1.287 radians. The tangent to \(C\) at \(P\) and the tangent to \(C\) at \(Q\) intersect on the \(y\)-axis at the point \(R\).
4. Find the area of the shaded region \(P Q R\) shown in Figure 2. \includegraphics[max width=\textwidth, alt={}, center]{e7043e7a-2c8f-425a-8471-f647828cc297-25_177_154_2576_1804}

8. \begin{figure}[h] \end{figure} Figure 2 shows the cross section \(A B C D\) of a small shed. The straight line \(A B\) is vertical and has length 2.12 m . The straight line \(A D\) is horizontal and has length 1.86 m . The curve \(B C\) is an arc of a circle with centre \(A\), and \(C D\) is a straight line. Given that the size of \(\angle B A C\) is 0.65 radians, find

1. the length of the arc \(B C\), in m , to 2 decimal places,
2. the area of the sector \(B A C\), in \(\mathrm { m } ^ { 2 }\), to 2 decimal places,
3. the size of \(\angle C A D\), in radians, to 2 decimal places,
4. the area of the cross section \(A B C D\) of the shed, in \(\mathrm { m } ^ { 2 }\), to 2 decimal places.

7. \begin{figure}[h] \end{figure} Figure 1 shows \(A B C\), a sector of a circle with centre \(A\) and radius 7 cm .
Given that the size of \(\angle B A C\) is exactly 0.8 radians, find

1. the length of the arc \(B C\),
2. the area of the sector \(A B C\). The point \(D\) is the mid-point of \(A C\). The region \(R\), shown shaded in Figure 1, is bounded by \(C D , D B\) and the arc \(B C\). Find
3. the perimeter of \(R\), giving your answer to 3 significant figures,
4. the area of \(R\), giving your answer to 3 significant figures.

9. \begin{figure}[h] \end{figure} Figure 2 shows a closed box used by a shop for packing pieces of cake. The box is a right prism of height \(h \mathrm {~cm}\). The cross section is a sector of a circle. The sector has radius \(r \mathrm {~cm}\) and angle 1 radian. The volume of the box is \(300 \mathrm {~cm} ^ { 3 }\).

1. Show that the surface area of the box, \(S \mathrm {~cm} ^ { 2 }\), is given by $$S = r ^ { 2 } + \frac { 1800 } { r }$$
2. Use calculus to find the value of \(r\) for which \(S\) is stationary.
3. Prove that this value of \(r\) gives a minimum value of \(S\).
4. Find, to the nearest \(\mathrm { cm } ^ { 2 }\), this minimum value of \(S\).

6. \includegraphics[max width=\textwidth, alt={}, center]{571780c2-945b-4636-b7c3-0bd558d28710-07_458_809_258_569} \section*{Figure 1} Figure 1 shows the sector \(O A B\) of a circle with centre \(O\), radius 9 cm and angle 0.7 radians.

1. Find the length of the arc \(A B\).
2. Find the area of the sector \(O A B\). The line \(A C\) shown in Figure 1 is perpendicular to \(O A\), and \(O B C\) is a straight line.
3. Find the length of \(A C\), giving your answer to 2 decimal places. The region \(H\) is bounded by the arc \(A B\) and the lines \(A C\) and \(C B\).
4. Find the area of \(H\), giving your answer to 2 decimal places.
   \section*{LU}

5. \begin{figure}[h] \end{figure} The shape shown in Figure 1 is a pattern for a pendant. It consists of a sector \(O A B\) of a circle centre \(O\), of radius 6 cm , and angle \(A O B = \frac { \pi } { 3 }\). The circle \(C\), inside the sector, touches the two straight edges, \(O A\) and \(O B\), and the \(\operatorname { arc } A B\) as shown. Find

1. the area of the sector \(O A B\),
2. the radius of the circle \(C\). The region outside the circle \(C\) and inside the sector \(O A B\) is shown shaded in Figure 1.
3. Find the area of the shaded region.

3. \begin{figure}[h] \end{figure} The circle \(C\) with centre \(T\) and radius \(r\) has equation $$x ^ { 2 } + y ^ { 2 } - 20 x - 16 y + 139 = 0$$

1. Find the coordinates of the centre of \(C\).
2. Show that \(r = 5\) The line \(L\) has equation \(x = 13\) and crosses \(C\) at the points \(P\) and \(Q\) as shown in Figure 1.
3. Find the \(y\) coordinate of \(P\) and the \(y\) coordinate of \(Q\). Given that, to 3 decimal places, the angle \(P T Q\) is 1.855 radians,
4. find the perimeter of the sector \(P T Q\).

8. \begin{figure}[h] \end{figure} Figure 2 shows the design for a triangular garden \(A B C\) where \(A B = 7 \mathrm {~m} , A C = 13 \mathrm {~m}\) and \(B C = 10 \mathrm {~m}\). Given that angle \(B A C = \theta\) radians,

1. show that, to 3 decimal places, \(\theta = 0.865\) The point \(D\) lies on \(A C\) such that \(B D\) is an arc of the circle centre \(A\), radius 7 m .
   The shaded region \(S\) is bounded by the arc \(B D\) and the lines \(B C\) and \(D C\). The shaded region \(S\) will be sown with grass seed, to make a lawned area. Given that 50 g of grass seed are needed for each square metre of lawn,
2. find the amount of grass seed needed, giving your answer to the nearest 10 g .

5. \begin{figure}[h] \end{figure} Figure 2 shows a plan view of a garden.
The plan of the garden \(A B C D E A\) consists of a triangle \(A B E\) joined to a sector \(B C D E\) of a circle with radius 12 m and centre \(B\).
The points \(A , B\) and \(C\) lie on a straight line with \(A B = 23 \mathrm {~m}\) and \(B C = 12 \mathrm {~m}\).
Given that the size of angle \(A B E\) is exactly 0.64 radians, find

1. the area of the garden, giving your answer in \(\mathrm { m } ^ { 2 }\), to 1 decimal place,
2. the perimeter of the garden, giving your answer in metres, to 1 decimal place.

5. \begin{figure}[h] \end{figure} Figure 2 shows the shape \(A B C D E A\) which consists of a right-angled triangle \(B C D\) joined to a sector \(A B D E A\) of a circle with radius 7 cm and centre \(B\). \(A , B\) and \(C\) lie on a straight line with \(A B = 7 \mathrm {~cm}\).
Given that the size of angle \(A B D\) is exactly 2.1 radians,

1. find, in cm, the length of the arc \(D E A\),
2. find, in cm, the perimeter of the shape \(A B C D E A\), giving your answer to 1 decimal place.

5. \begin{figure}[h] \end{figure} The shape \(A B C D E A\), as shown in Figure 2, consists of a right-angled triangle \(E A B\) and a triangle \(D B C\) joined to a sector \(B D E\) of a circle with radius 5 cm and centre \(B\). The points \(A , B\) and \(C\) lie on a straight line with \(B C = 7.5 \mathrm {~cm}\).
Angle \(E A B = \frac { \pi } { 2 }\) radians, angle \(E B D = 1.4\) radians and \(C D = 6.1 \mathrm {~cm}\).

1. Find, in \(\mathrm { cm } ^ { 2 }\), the area of the sector \(B D E\).
2. Find the size of the angle \(D B C\), giving your answer in radians to 3 decimal places.
3. Find, in \(\mathrm { cm } ^ { 2 }\), the area of the shape \(A B C D E A\), giving your answer to 3 significant figures.

4. \begin{figure}[h] \end{figure} Figure 1 shows a sketch of a design for a scraper blade. The blade \(A O B C D A\) consists of an isosceles triangle \(C O D\) joined along its equal sides to sectors \(O B C\) and \(O D A\) of a circle with centre \(O\) and radius 8 cm . Angles \(A O D\) and \(B O C\) are equal. \(A O B\) is a straight line and is parallel to the line \(D C . D C\) has length 7 cm .

1. Show that the angle \(C O D\) is 0.906 radians, correct to 3 significant figures.
2. Find the perimeter of \(A O B C D A\), giving your answer to 3 significant figures.
3. Find the area of \(A O B C D A\), giving your answer to 3 significant figures.

9. \begin{figure}[h] \end{figure} Figure 4 shows a plan view of a sheep enclosure.
The enclosure \(A B C D E A\), as shown in Figure 4, consists of a rectangle \(B C D E\) joined to an equilateral triangle \(B F A\) and a sector \(F E A\) of a circle with radius \(x\) metres and centre \(F\). The points \(B , F\) and \(E\) lie on a straight line with \(F E = x\) metres and \(10 \leqslant x \leqslant 25\)

1. Find, in \(\mathrm { m } ^ { 2 }\), the exact area of the sector \(F E A\), giving your answer in terms of \(x\), in its simplest form. Given that \(B C = y\) metres, where \(y > 0\), and the area of the enclosure is \(1000 \mathrm {~m} ^ { 2 }\),
2. show that $$y = \frac { 500 } { x } - \frac { x } { 24 } ( 4 \pi + 3 \sqrt { 3 } )$$
3. Hence show that the perimeter \(P\) metres of the enclosure is given by $$P = \frac { 1000 } { x } + \frac { x } { 12 } ( 4 \pi + 36 - 3 \sqrt { 3 } )$$
4. Use calculus to find the minimum value of \(P\), giving your answer to the nearest metre.
5. Justify, by further differentiation, that the value of \(P\) you have found is a minimum.

4. \begin{figure}[h] \end{figure} Diagram not drawn to scale Figure 1 is a sketch representing the cross-section of a large tent \(A B C D E F\). \(A B\) and \(D E\) are line segments of equal length.
Angle \(F A B\) and angle \(D E F\) are equal. \(F\) is the midpoint of the straight line \(A E\) and \(F C\) is perpendicular to \(A E\). \(B C D\) is an arc of a circle of radius 3.5 m with centre at \(F\).
It is given that $$\begin{aligned} A F & = F E = 3.7 \mathrm {~m} \\ B F & = F D = 3.5 \mathrm {~m} \\ \text { angle } B F D & = 1.77 \text { radians } \end{aligned}$$ Find

1. the length of the arc \(B C D\) in metres to 2 decimal places,
2. the area of the sector \(F B C D\) in \(\mathrm { m } ^ { 2 }\) to 2 decimal places,
3. the total area of the cross-section of the tent in \(\mathrm { m } ^ { 2 }\) to 2 decimal places.