| 21 | $( 2018/s/41/img/q6) $ A solid hemisphere has volume \(230 \, \text{cm}^3\). (a) Calculate the radius of the hemisphere. [The volume, \(V\), of a sphere with radius \(r\) is \(V = \frac{4}{3}\pi r^3\).] [3] (b) A solid cylinder with radius \(1.6 \, \text{cm}\) is attached to the hemisphere to make a toy.  The total volume of the toy is \(300 \, \text{cm}^3\). (i) Calculate the height of the cylinder.[3] (ii) A mathematically similar toy has volume \(19200 \, \text{cm}^3\). Calculate the radius of the cylinder for this toy.[3] |
| 22 | $( 2018/w/41/img/q5) $ The diagram shows a prism with length \(18 \, \text{cm}\) and volume \(253.8 \, \text{cm}^3\). The cross-section of the prism is a right-angled triangle with base \(6 \, \text{cm}\) and height \(h \, \text{cm}\). (a) (i) Show that the value of \(h\) is \(4.7\). (ii) Calculate the value of \(x\).[3] (b) Calculate the total surface area of the prism.[6] |
| 23 | $( 2018/s/21/img/q6) $ (a)A sphere of radius \(r\) is inside a closed cylinder of radius \(r\) and height \(2r\). [The volume, \(V\), of a sphere with radius \(r\) is \(V = \frac{4}{3}\pi r^3\).] (i) When \(r = 8 \, \text{cm}\), calculate the volume inside the cylinder which is \textbf{not} occupied by the sphere.[3] (ii) Find \(r\) when the volume inside the cylinder \textbf{not} occupied by the sphere is \(36 \, \text{cm}^3\).[3] (b)  The diagram shows a solid cone with radius \(5 \, \text{cm}\) and perpendicular height \(12 \, \text{cm}\). (i) The total surface area is painted at a cost of \(\$0.015\) per \(\text{cm}^2\). Calculate the cost of painting the cone. [The curved surface area, \(A\), of a cone with radius \(r\) and slant height \(l\) is \(A = \pi r l\).][4] (ii) The cone is made of metal and is melted down and made into smaller solid cones with radius \(1.25 \, \text{cm}\) and perpendicular height \(3 \, \text{cm}\). Calculate the number of smaller cones that can be made.[3] |
| 24 | $( 2018/m/42/img/q5) $ (a)  The diagram shows a solid prism with length \(15.2 \, \text{cm}\). The cross-section of this prism is a regular hexagon with side \(7 \, \text{cm}\). (i) Calculate the volume of the prism.[5] (ii) Calculate the total surface area of the prism.[3] (b) Another solid metal prism with volume \(500 \, \text{cm}^3\) is melted and made into \(6\) identical spheres. Calculate the radius of each sphere.[3] [The volume, \(V\), of a sphere with radius \(r\) is \(V = \frac{4}{3}\pi r^3\).] |
| 25 | $( 2017/s/43/img/q4) $ The diagram shows a solid metal prism with cross section \(ABCDE\)  (i) Calculate the area of the cross section \(ABCDE\).[6] (ii) The prism is of length \(8\ \text{cm}\). Calculate the volume of the prism.[1] (b) A cylinder of length \(13\ \text{cm}\) has volume \(280\ \text{cm}^3\). (i) Calculate the radius of the cylinder.[3] (ii) The cylinder is placed in a box that is a cube of side \(14\ \text{cm}\). Calculate the percentage of the volume of the box that is occupied by the cylinder.[3] |
| 26 | $( 2017/w/43/img/q6) $ The diagrams show a cube, a cylinder and a hemisphere. The volume of each of these solids is \(2000\ \text{cm}^3\). (i) Work out the height, \(h\), of the cylinder.[2] (ii) Work out the radius, \(r\), of the hemisphere.[3] \[ \text{[The volume, } V \text{, of a sphere with radius } r \text{ is } V = \frac{4}{3}\pi r^3.\text{]} \] (iii) Work out the surface area of the cube.[3]  (b)(i) Calculate the area of the triangle.[2] (ii) Calculate the perimeter of the triangle and show that it is \(23.5\ \text{cm}\), correct to one decimal place. Show all your working.[5]  (c) The perimeter of this sector of a circle is \(28.2\ \text{cm}\). Calculate the value of \(c\).[3] |
| 27 | $( 2018/s/21/img/q6) $ (a)  Water flows through a cylindrical pipe at a speed of \(8\ \text{cm/s}\). The radius of the circular cross-section is \(1.5\ \text{cm}\) and the pipe is always completely full of water. Calculate the amount of water that flows through the pipe in 1 hour. Give your answer in litres.[4] (b)  The diagram shows three solids. The base radius of the cone is \(6\ \text{cm}\) and the slant height is \(12\ \text{cm}\). The radius of the sphere is \(x\ \text{cm}\) and the radius of the hemisphere is \(y\ \text{cm}\). The total surface area of each solid is the same. (i) Show that the total surface area of the cone is \(108\pi\ \text{cm}^2\).[2] \[ \text{[The curved surface area, } A \text{, of a cone with radius } r \text{ and slant height } l \text{ is } A = \pi rl.\text{]} \] (ii)] Find the value of \(x\) and the value of \(y\).[4] \[ \text{[The surface area, } A \text{, of a sphere with radius } r \text{ is } A = 4\pi r^2.\text{]} \] |
| 28 | $( 2018/w/22/img/q21) $ (a)  The diagram shows a solid cone. The radius is \(8\ \text{cm}\) and the slant height is \(17\ \text{cm}\). \ (i) Calculate the curved surface area of the cone.[2] \[ \text{[The curved surface area, } A \text{, of a cone with radius } r \text{ and slant height } l \text{ is } A = \pi rl.\text{]} \] (ii) Calculate the volume of the cone.[4] \[ \text{[The volume, } V \text{, of a cone with radius } r \text{ and height } h \text{ is } V = \frac{1}{3}\pi r^2 h.\text{]} \] (iii) The cone is made of wood and \(1\ \text{cm}^3\) of the wood has a mass of \(0.8\ \text{g}\). Calculate the mass of the cone.[3] (iv) The cone is placed in a box. The total mass of the cone and the box is \(1.2\ \text{kg}\). Calculate the mass of the box. Give your answer in grams.[1]  The diagram shows a solid cylinder and a solid sphere. The cylinder has radius \(3r\) and height \(8r\). The sphere has radius \(r\). (i) Find the volume of the sphere as a fraction of the volume of the cylinder. Give your answer in its lowest terms.[4] \[ \text{[The volume, } V \text{, of a sphere with radius } r \text{ is } V = \frac{4}{3}\pi r^3.\text{]} \] (ii) The surface area of the sphere is \(8\pi\ \text{cm}^2\). Find the \textbf{curved} surface area of the cylinder. Give your answer in terms of \(\pi\).[4] \[ \text{[The surface area, } A \text{, of a sphere with radius } r \text{ is } A = 4\pi r^2.\text{]} \] |
Answer Key
21. (a) 4.79 or 4.788 to 4.789 (b)(i) \( 8.7[0] \) or 8.702 to 8.704 (ii) 6.4
22. (a) (i) Show (a) (ii) 38.1 or 38.06 to 38.08 (b) 358 or 357.9 to 358
23. (a) (i) 1070 or 1072 . .. (ii) 2.58 or 2.580 to 2.581 (b) (i) 4.24 or 4.241 to 4.242 (ii) 64
24. (a) (i) 1930 or 1940 or 1933.4 to 1935.3 (ii) 893 or 892.8 to \( 893.0 \ldots \) (b) 2.71 or 2.709 to 2.710
25. (a)(i) -1.33 < k < 0 to 0.1 ( a) 17.5 or \( 17.46 \ldots \) (a) (ii) 140 or 139.6 to \( 139.7 \ldots \) (b) (i) 2.62 or \( 2.618 \ldots \) (b) (ii) 10.2 or \( 10.20 \ldots 10 \frac{10}{49} \)
26. (a)(i)25.5 or 25.46 (a)(ii)9.85 or 9.847 (a) (iii) 952 or 952.4 (b)(i) 22.5 or 22.49 (b) (ii) 23.46 (c) 64.9 or 64.92 to 64.94
27. (a) 204 or 203.5 to 203.6 (b) (i) Show (b) (ii)[x \( =] 5.2[0] \) or \( 5.196[\mathrm{y}=] 6 \)
28. (a) (i) 427 or 427.2 to 427.3 (a) (ii) 1010 or 1005 (a) (iii) 804 or 804.2 to 804.4 or 808
(a) (iv) 396 or 395.6 to 395.8 or 392 (b) (i) \( \frac{1}{54} \) (b) (ii) \( 972 \pi \)
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