A connection which could be illustrated, and could be understood by students who know the perimeter of a circle, runs as follows: Put the sphere of radius R inside a cylinder, with the cylinder just touching the equator, and cut off at the height of the top and bottom of the sphere. (A cutaway view is in the diagram.) [pic]

What is the area of the curved part of the cylinder? 2 Pi R x 2R = 4 Pi R2. This is found by slicing the cylinder surface and rolling it out as a rectangle. Now, it is NOT an accident that the cylinder surface is EXACTLY the area of the sphere. Take in small horizontal slice through the diagram. (I have colored one such slice orange.) This cuts a rectangle out of the rolled out cylinder and slightly distorted rectangle out of the sphere. (If the slice is very thin then the distortion is "slight".) In the cross-sectional view below hc is the height of the slice on the cylinder, hs is the length of the arc on the sphere cut out by the slice, r is the radius of the distorted rectangle on the sphere and R is the radius of the sphere. [pic]

The area of the orange rectangle on the cylinder is 2 Pi R hc and the area of the distorted orange rectangle on the sphere is approximately 2 Pi r hs. These two areas are approximately equal (the proof is outlined below). Since the cylinder and the sphere can be decomposed into these rectangular strips, the area of the sphere and the area of the cylinder are approximately equal. This argument is in the spirit of how the Greeks compared slices to show that areas and volumes are the same. It is not only interesting reasoning in proportion, etc., but it is a lesson in History. In the same spirit, you can compare the VOLUME of a hemi-sphere with that of the cylinder with an inverted cone removed. (I think of the hemisphere with the equator at the bottom and the cone...

...SurfaceArea Formulas
In general, the surfacearea is the sum of all the areas of all the shapes that cover the surface of the object.
Cube | Rectangular Prism | Prism | Sphere | Cylinder | Units
Note: "ab" means "a" multiplied by "b". "a2" means "a squared", which is the same as "a" times "a".
Be careful!! Units count. Use the same units for all measurements. Examples
|SurfaceArea of a Cube = 6 a 2 |
[pic](a is the length of the side of each edge of the cube)
In words, the surfacearea of a cube is the area of the six squares that cover it. The area of one of them is a*a, or a 2 . Since these are all the same, you can multiply one of them by six, so the surfacearea of a cube is 6 times one of the sides squared.
|SurfaceArea of a Rectangular Prism = 2ab + 2bc + 2ac |
[pic](a, b, and c are the lengths of the 3 sides)
In words, the surfacearea of a rectangular prism is the are of the six rectangles that cover it. But we don't have to figure out all six because we know that the top and bottom are the same, the front...

...RIGHT
Find the area of a square with a side length of 4 inches....C
Find the volume of concrete needed to make a circular patio that has a radius of 24 feet and is 8 feet thick. Use 3.14 for pi. If necessary, round the answer to the nearest cubic foot....C
B
Write the equation of the circle with the given center and radius. Then graph the circle.
center: (–1, –3) radius: 6.....B
A
The volumes of 2 similar solids are 27 and 125 The surfacearea of the larger solid is 250 What is the surfacearea of the smaller solid? Round your answer to the tenths place....C
A regular nonagon has a side length of feet. A similar nonagon has a side length of 5 feet. What is the ratio of the first nonagon's perimeter to the second nonagon's perimeter? What is the ratio of the first nonagon's area to the second nonagon's area?C
A polyhedron has 20 faces, and all of them are triangles. If the polyhedron has 30 edges, how many vertices does it have?B
A circle has an area of 153.86 square inches. Using 3.14 for pi, find the radius and circumference of the circle. Round your answer to the nearest hundredth when necessary.....A
C
D
What is the shape of the intersection of a sphere and a horizontal plane?D
A
A
You are ordering your school pictures. You decide to order one 8-by-10-inch, two 5-by-7-inch, and four 2.5-by-3.5-inch photos. Are...

...SurfaceareaSurfacearea is the measure of how much exposed area a solid object has, expressed in square units. Mathematical description of the surfacearea is considerably more involved than the definition of arc length of a curve. For polyhedra (objects with flat polygonal faces) the surfacearea is the sum of the areas of its faces. Smoothsurfaces, such as a sphere, are assigned surfacearea using their representation as parametric surfaces. This definition of the surfacearea is based on methods of infinitesimal calculus and involves partial derivatives and double integration.
General definition of surfacearea was sought by Henri Lebesgue and Hermann Minkowski at the turn of the twentieth century. Their work led to the development of geometric measure theory which studies various notions of surfacearea for irregular objects of any dimension. An important example is the Minkowski content of a surface.
Definition of surfacearea
While areas of many simple surfaces have been known since antiquity, a rigorous mathematical definition of area requires a lot of care. Surface...

...corresponding base. If the altitude were increased by
4cm and the base decreased by 2cm, the area of the triangle would remain the same. Find the base and altitude of the triangle.
2. Some toffees are bought at the rate of 11 for Rs10 and same numbers are bought at the
rate of 9 for Rs 10. If the whole lot is sold at one rupee per toffee, find the gain or loss percent.
3. Chandu purchased a watch at 20% discount on its marked price but sold it at marked price.
Find the gain percent of Chandu on this transaction.
4. A motorboat covers a certain distance downstream in a river in five hours. It covers the same distance upstream in five hours and half. The speed of water is 1.5 km/hr. Find the speed of the boat in still water.
5. Factorize:
(i) 2x2+y2+8z3-22xy-42yz+8xz
(ii) x6 – 3x4y2 +3x2y4 –y6
6. Evaluate: (367/2 –369/2)/ 365/2
7. Divide 34x-22x3-12x4-10x2-75 by (3x+7) and check your answer.
8. The digit in the tens place of a number is three times that in the ones place. If the digits are reversed, the new number will be 36 less than the original number. Find the number.
9. A well is dug 20m deep and it has a diameter 7m. The earth, which is so dug out, is spread out on a rectangular plot 22m long and 14m broad. What is the height of the platform so formed?
10. The total surfacearea of a hollow cylinder open at both ends is 4620sqcm, area of the base ring is...

...F in half an hour. Show that its temperature will be 81.6◦ F after 3 hours
of cooling.
3. In very dry regions, the phenomenon called Virga is very important because it can endanger
aeroplanes. [See http://en.wikipedia.org/wiki/Virga ]. Virga is rain in air that is so dry that the
raindrops evaporate before they can reach the ground. Suppose that the volume of a raindrop is
proportional to the 3/2 power of its surfacearea. [Why is this reasonable? Note: raindrops are not
spherical, but let’s assume that they always have the same shape, no matter what their size may
be.] Suppose that the rate of reduction of the volume of a raindrop is proportional to its surfacearea. [Why is this reasonable?] Find a formula for the amount of time it takes for a virga raindrop
to evaporate completely, expressed in terms of the constants you introduced and the initial surfacearea of a raindrop. Check that the units of your formula are correct. Suppose somebody suggests
that the rate of reduction of the volume of a raindrop is proportional to the square of the surfacearea. Argue that this cannot be correct.
4. One theory about the behaviour of moths states that they navigate at night by keeping a
ﬁxed angle between their velocity vector and the direction of the Moon [or some bright star; see
http://en.wikipedia.org/wiki/Moth]. A certain moth ﬂies near to a candle...

... SECTION A (40 Marks)
Attempt all questions from this Section
Question 1.
a) What number must be subtracted from 2x3 – 5x2 + 5x so that the resulting polynomial has a factor 2x – 3 ? [3]
b) D, E, F are mid points of the sides BC, CA and AB respectively of a Δ ABC. Find the ratio of the areas of Δ DEF and Δ ABC. [3]
c) A man borrowed a sum of money and agrees to pay off by paying Rs 3150 at the end of the first year and Rs 4410 at the end of the second year. If the rate of compound interest is 5% per annum, find the sum borrowed. [4]
Question 2.
a) The y-axis is a line of symmetry for the figure ACBD where A, B have co-ordinates (3, 6), (– 3, 4) respectively. (i) Find the co-ordinates of C and D. (ii) Name the figure ACBD and find its area. [3]
b) PAQ is a tangent at A to the circumcircle of Δ ABC such that PAQ is parallel to BC, prove that ABC is an isosceles triangle. [3]
c) A rectangular piece of paper 30 cm long and 21 cm wide is taken. Find the area of the biggest circle that can be cut out from this paper. Also find the area of the paper left after cutting out the circle. [Take π = 22/7] [4]
Question 3.
a) Construct a 2 × 2 matrix whose elements aij are given by aij = i + j. [3]
b) The point P (– 4, – 5) on reflection in y-axis is mapped on P’. The point P’ on reflection...

...Mathematics
Volume of Solids
Formulae for Volume of Solids
Cube | Cuboid | Triangular Prism | Cylinder | Cone | Pyramid | Sphere | AnyPrism |
s3 | lwh | ½bhl | Πr2h | 1/3πr2h | 1/3Ah | 4/3πr3 | Ah |
A = area of the base of the figure
s = length of a side of the figure
l = length of the figure
w = width of the figure
h = height of the figure
π = 22/7 or 3.14
1. Compute the volume of a cube with side 7cm.
Volume of cube: s3
s = 7cm
s3 = (7cm x 7cm x 7cm) = 343cm3
2. Compute the volume of a cuboid (also known as a rectangular prism) with the dimensions 4cm by 13cm by 9cm.
Volume of a cuboid: lwh
l = 4cm
w = 13cm
h = 9cm
lwh = (4cm x 13cm x 9cm) = 468cm3
3. Compute the volume of a triangular prism with a base length of 60cm, a base width of 8cm, and a height of 10cm.
Volume of a triangular prism: ½bhl
½b = (8cm x 1/2) = 4cm
h = 10cm
l = 60cm
½bhl = (4cm x 10cm x 60cm) = 2400cm3
4. Compute the volume of a cylinder which is 2m tall and has a radius 75cm. Convert this litres.
Volume of a cylinder: πr2h
π = 3.1415
r2 = (75cm)2 = 375 cm2
h = 2m = 200cm
πr2h = 235612. 5 cm3
cm L = 1cm 0.001 L
235612.5 cm3/ 1000 = 235.6125 L
5. Compute the volume of a cone with a radius of 200cm and a height of 0.75m.
Volume of a cone: 1/3πr2h
1/3π = 1.047
r2 = (2m)2 = 4m2
h = 0.75m
1/3πr2h = (1.047 x 0.75m x 4m2) = 3.141 m3
6. Compute the volume of pyramid with a base length of 10cm, base width 10cm, and...

...Intro: SurfaceArea and Volume
Multiple Choice Identify the choice that best completes the statement or answers the question. Find the surfacearea of the space figure represented by the net. ____ 1.
12 in. 4 in.
6 in.
4 in. 4 in. 6 in.
a. 288 in.2 ____ 2.
b. 144 in.2
c. 240 in.2
d. 288 in.2
5 cm 5 cm
7 cm 8 cm 4 cm
____
a. 124 cm2 b. 110 cm2 c. 150 cm2 d. 164 cm2 3. Find the surfacearea of the cylinder. Use a calculator. Round to the nearest tenth.
4m
3m
a. 125.7 m2
b. 138.2 m2
c. 150.8 m2
d. 175.9 m2
____
4. Find the surfacearea of a cylinder with radius 5.9 ft and height 4.4 ft. Use a calculator. Round to the nearest tenth. a. 300.3 ft2 b. 481.2 ft2 c. 381.8 ft2 d. 272.5 ft2 5. Find the surfacearea of the square pyramid.
____
4 ft
8 ft Diagram not to scale.
a. 64 ft2 ____
b. 128 ft2
c. 80 ft2
d. 96 ft2
6. Find the surfacearea of a cone with radius of 8.8 cm and slant height of 7.1 cm, to the nearest square unit. Use 3.14 for π. a. 636 cm2 b. 341 cm2 c. 440 cm2 d. 683 cm2 7. Find the surfacearea of the cone to the nearest square unit. Use π = 3.14.
____
3 cm
6 cm Diagram not to scale.
a. 283 cm2 ____
b. 170 cm2
c. 141 cm2
d. 226 cm2
8. George made a conical hat to match...

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