Surface area
Surface area is the measure of how much exposed area a solid object has, expressed in square units. Mathematical description of the surface area is considerably more involved than the definition of arc length of a curve. For polyhedra (objects with flat polygonal faces) the surface area is the sum of the areas of its faces. Smooth surfaces, such as a sphere, are assigned surface area using their representation as parametric surfaces. This definition of the surface area is based on methods of infinitesimal calculus and involves partial derivatives and double integration. General definition of surface area 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 surface area for irregular objects of any dimension. An important example is the Minkowski content of a surface.

Definition of surface area

While areas of many simple surfaces have been known since antiquity, a rigorous mathematical definition of area requires a lot of care. Surface area is an assignment of a positive real number to a certain class of surfaces that satisfies several natural requirements. The most fundamental property of the surface area is its additivity: the area of the whole is the sum of the areas of the parts. More rigorously, if a surface S is a union of finitely many pieces S1, …, Sr which do not overlap except at their boundaries then

Surface areas of flat polygonal shapes must agree with their geometrically defined area. Since surface area is a geometric notion, areas of congruent surfaces must be the same and area must depend only on the shape of the surface, but not on its position and orientation in space. This means that surface area is invariant under the group of Euclidean motions. These properties uniquely characterize surface area for a wide class of geometric surfaces called piecewise smooth. Such surfaces consist of finitely...

...Planning
Aim
To determine how the surfacearea of the tablets affects the rate of the reaction. To determine which form of tablets gives the biggest surfacearea resulting in the fastest reaction rate.
Investigation question:
What is the relationship between the total surfacearea of the tablets and the rate of the reaction?
Hypothesis:
The rate of reaction will be the fastest when the tablets crushed into powder as there is a bigger total surfacearea resulting in more effective collisions between particles.
Variables:
Independent variable: Different forms of tablets.
Dependant variable: Time the syringe took to stop moving as the tablets dissolve.
Fixed variables:
*External temperature
*volume of HCl
*Temperature –all 3 final runs were done on the same day so whether was not an issue and did not affect the results
*Use of catalyst – a catalyst was not used in any of the experiments
* Use the same person to observe the reaction because different people have different eyesight
Background information relating to the experiment
In this experiment we are looking at one effect that influences the rate of reaction , namely total surfacearea. The reaction rate (rate of reaction) or speed of reaction for a reactant or product in a particular reaction is defined as how fast or slow a reaction takes place....

...Surfacearea / Volume ratio Experiment
Introduction:
The surfacearea to volume ratio in living organisms is very important. Nutrients and oxygen need to diffuse through the cell membrane and into the cells. Most cells are no longer than 1mm in diameter because small cells enable nutrients and oxygen to diffuse into the cell quickly and allow waste to diffuse out of the cell quickly. If the cells were any bigger than this then it would take too long for the nutrients and oxygen to diffuse into the cell so the cell would probably not survive. Single celled organisms can survive as they have a large enough surfacearea to allow all the oxygen and nutrients they need to diffuse through. Larger multi celled organisms need specialist organs to respire such as lungs or gills.
Apparatus Needed For the Experiments:
1. Beakers
2. Gelatin blocks mixed containing universal indicator
3. 0.1M Hydrochloric acid
4. Stop Watch
5. Scalpel
6. Tile
7. Safety glasses
Method:
1. A block of gelatin which has been dyed with universal indicator should be cut into blocks of the following sizes (mm).
5 x 5 x 5
10 x 10 x 10
15 x 15 x 15
20 x 20 x 20
10 x 10 x 2
10 x 10 x 10 (Triangle)
10 x 15 x 5
20 x 5 x 5
The triangle is of the following dimensions.
The rest of the blocks are just plain cubes or rectangular blocks.
Universal indicator is a neutral indicator....

...SurfaceArea to Volume Ratio and the Relation to the Rate of Diffusion
Aim and Background
This is an experiment to examine how the SurfaceArea / Volume Ratio affects the rate of diffusion and how this relates to the size and shape of living organisms.
The surfacearea to volume ratio in living organisms is very important. Nutrients and oxygen need to diffuse through the cell membrane and into the cells. Most cells are no longer than 1mm in diameter because small cells enable nutrients and oxygen to diffuse into the cell quickly and allow waste to diffuse out of the cell quickly. If the cells were any bigger than this then it would take too long for the nutrients and oxygen to diffuse into the cell so the cell would probably not survive.
Single celled organisms can survive as they have a large enough surfacearea to allow all the oxygen and nutrients they need to diffuse through. Larger multi-celled organisms need organs to respire such as lungs or gills.
Method
The reason I chose to do this particular experiment is because I found it very interesting and also because the aim, method, results- basically the whole experiment would be easily understood by the average person who knew nothing about SurfaceArea/Volume Ratio. The variable being tested in this experiment is the rate of diffusion in relation to the size of the gelatin...

...AREA
(i) The area of a rhombus is equal to the area of a triangle whose base and the corresponding altitude are 24.8 cm and 16.5 cm respectively. If one of the diagonal of the rhombus is 22 cm, find the length of the other diagonal.
(ii) The floor of a rectangular hall has a perimeter 250m. If the cost of paining the four walls at the rate of Rs 10 per m2 is Rs 1500. Find the height of the hall.
(iii) A room is half as long again as it is broad. The cost of carpeting the room at Rs 3.25 per m2 is Rs 175.50 and the cost of papering the walls at Rs 1.40 per m2 is Rs 240.80. If 1 door and 2 windows occupy 8m2, find the dimensions of the room.
(iv) A river 2m deep and 45m wide is flowing at the rate of 3 km per hour. Find the volume of water that runs into the sea per minute.
(v) A closed cylinder has diameter 8cm and height 10cm. Find its total surfacearea and volume.
(vi) The volume of a metallic cylinder pipe is 748cm3 . Its length is 14 cm and external diameter 18cm. Find its thickness.
(vii) A cylindrical bucket, 28cm in diameter 72cm high is full of water. The water is emptied into a rectangular tank, 66cm long and 28cm wide. Find the height of the water level in the tank.
(viii) A cylindrical tube, open at both ends, is made of metal. The internal diameter of the tube is 10.4cm and its length is 25cm. The thickness of the metal is 8mm...

...Kimberly M Dollar
000234333
EFT4: Math: Task 5: SurfaceArea of Cubes
Introducing SurfaceArea
For a fifth or sixth grade class to understand the concept surfacearea in relation to a cube they need to understand what a cube is first. They will learn that a cube is a special type of rectangular solid. The length, width, and height of a cube are exactly the same. After explaining what a cube is they will need to understand what it means to find the surfacearea. The surfacearea is not the same as finding the volume of a cube. The surfacearea is the area on the outside of a three-dimensional shape, like the cube. The surfacearea of a cube is six times the surfacearea of one side of the cube. There are six sides to one cube, after learning this about a cube the appropriate formula to find the surfacearea is:
Surfacearea of a cube=6s^2 (The “6” represents the number of sides; “s” represents one side of a cube; “^2” represents taking one side and timing it by itself; the end result gives the surfacearea of a cube).
Prerequisite Skills
The necessary prerequisite skills required to determine the surfacearea...

...w
The Effect of Changing the surfacearea on the Rate of Reaction?
By :
22/10/2013
The effect of changing the surfacearea on the
rate of reaction?
Unit Question: Should we speed things up or slow them down?
Hypothesis:
According to collision theory, should the surfacearea increase the amount of collisions increase increasing the rate of reaction. Therefore, my hypothesis is that when the surfacearea increases the rate of reaction increases.
Background information:
“Collision theory is a theory proposed independently by Max Trautzin 1916 and William Lewis in 1918, that qualitatively explains how chemical reactions occur and why reaction rates differ for different reactions. The collision theory states that when suitable particles of the reactant hit each other, only a certain percentage of the collisions cause any noticeable or significant chemical change; these successful changes are called successful collisions. The successful collisions have enough energy, also known as activation energy, at the moment of impact to break the pre-existing bonds and form all new bonds. This results in the products of the reaction. Increasing the concentration of the reactant particles or raising the temperature, thus bringing about more collisions and therefore many more...

... 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...

...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...